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Statistics Questions and Answers March 07, 2024

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Derive the Moment Generating Function of a multivariate normal distribution.,Suppose you have the data on scores (in a 100point scale) given by 20 students of school A, 20 students of school B and 15 students of school C regarding the COVID protocol followed in the schools. How will you test whether the median score of the three schools differ
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Consider the following model Y B B X U Where Y Average wage of ith firm, X Number of employees. Suppose it is known from some previous survey that o 02X. How will you proceed with problem of estimating the model parameters and why
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Let A1, A2, Ag be independent events so that P(A)q, 0
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23 13 P(x) Suggest a probability mass function of a random variable X, where PX Find E(X). State and prove Chebychevs inequality. and E(X)1.
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. Derive the Moment Generating Function of a multivariate normal distribution.

Using Venn diagram show that (AUB) AB, where AC complement of A.
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A standard normal distribution has the following characteristics (A) the mean is equal to 0 and the variance is equal to 1 B the mean and the variance are both equal to 1 the mean is equal to the variance D the mean and the variance are both equal to 0

Construct a truth table for each given expression (20 Points). a) (a V CAb) a b) (c a b) (a v bb c)

Suppose that on an average 1 customer per minute arrive at a shop. What is the probability that the shopkeeper will wait more than 5 minutes before (i) both of the first two customers arrive, and (ii) the first customer arrive Assume that waiting times follows gamma distribution.

The preliminary survey results just came back on the specialty catalog project, and they look great The average planned order size was 42.33, well above the 15 that was hoped for. The group leader will probably be delightedafter all, 42.33 for each of the 1,300,000 target addresses comes out to over 55 million in average sales As part of the

3. During the first half of 1987, the average first mortgage granted by U.S, credit unions was 52.026. Assuming a normal distribution and a standard deviation of 10,000, what proportion of the mortgages were a.Less than 52,026 b.between 32, 026 and 72,026

A maker of a certain brand of low-fat cereal bars claims that the averagesaturated fat content is 0.5 gram. In a random sample of 8 cereal bars ofthis brand, the saturated fat content was 0.6, 0.7, 0.7, 0.3, 0.4, 0.5, 0.4, and0.2. Would you agree the claim? Assume a normal distribution.

Question

Game 1 You are playing a game where you get n turns. Each of your turns involves flipping a coin a number of times. On the first turn, you have 1 flip, on the second turn you have two flips, and so on until your nth turn when you flip the coin n times. All the flips are mutually independent. The coin you are using is biased to flip Heads with

4 The midterm grades on a chemistry exam, graded on a scale of 0 to 100, were 62, 64, 65, 65, 68, 70, 72, 72, 74, 75, 75, 75, 76, 78, 78, 81, 83, 83, 84, 85, 87, 88, 92, 95, 98, 98, 100, 100, 740 (enoitgo lia) a) Using a graph, identify any outliers in this data If the outliers exist, how would you address the situation (8 marks) OITEATER b)
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Answer any ten questions (a) If we throw two fair dice, which is more likely a sum of 11 or a sum of 12 (b) Give a combinatorial explanation of the identity (3)(8) (E) (8)(7) (2)(6). (c) What is the frequentist view of probability (d) Give an example of a random experiment and two events A and B so that P(AB) P(BA). (e) State Bayes theorem. (f)
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P 1. (a) Let XXX, be p jointly distributed random variables. Derive the multiple correlation coefficient of X on X. X. X, . What will be the expression of this correlation coefficient when the true regression of X on X. X. linear (a) Let X. X. X, (k2) jointly follow a multinomial distribution with parameters (n. P1 P2 P ). Derive the marginal
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(a) Discuss the method of determination of probability that at least seven of the events A1, A2, A3, A20 occur simultaneously. State a problem where you can use the above result. (b) State and prove Bayes theorem. Construct a problem with four events E, F, F2 and F3, where F, F2 and F3 are mutually exclusive and exhaustive. Solve the problem using
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(a) State classical definition of probability and clearly mention its limitations. (b) If six married couples are seated at random at a round table, find the probability that no wife sits next to her husband. (c) Define independence of three events A, B and C. If A, B and C are independent, verify whether (i) B and A C and (ii) B and AC are
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(a) Derive the method of finding the probability that exactly m among N events A1, A2, ... AN occur simultaneously. Construct a problem and find the above probability for m3 and N5. (b) Define cumulative distribution function (cdf) and probability mass function (pmf) of a random variable. Suppose F(x)0 if x 0 if 0 x 11 1 if x 11, where x
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(a) Suppose you have one fair coin, and one biased coin which lands Heads with probability 45. You choose one of the coins at random and throw it three times independently. If it lands Heads all the three times, find the probability that the coin you picked is the fair one and comment. (b) Suppose coin R gives heads with probability and coin B
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1. State Leibnitz theorem. If log y tanx, then show that (1x)yn2(2nx2x1)y(nn)y 0 14 122 14 5 5 5 2. State Rolles theorem. Is the theorem true when the function is not continuous at the end points Justify your answer. Verify Rolles theorem for f(x)x25x6 on 2,3. lim 3. State LHospitals rule. Evaluate x 1(2x) tan 4. Find the asymptotes of the curve
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10. Find the condition that the line lx my n 0 may be a normal to the ellipse x y2 1 a 62 11. Show that the pair of tangents drawn from the center of a hyperbola are its asymptotes. 12. Solve the differential equation dy y tan dx X 13. Solve y2pxayp0 where p dy dx dy 14. Solve the differential equation xylogyxye dx 15. Solve the differential
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1. If y log(x a x), then show that (a x) yn2 (2n1)xy1 y 0 2. State and prove Lagranges Mean Value theorem. lim 3. Evaluate x II (sinx) tan x 4. Find the asymption of the curve ayxyax 2axx0 5. Find the radius of curvature at the origin for the curve x3 y 3axy 6. Evaluate xax dx
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11. Solve xp22yp ax 0 where p dy dx 12. Solve d2y dy 2 dx yxe dx 13. Solve x2 dx dy y log x 2 dx dx 14. Transform the equation x2xy yx3y0 to axes through the point (1,0) parallel to the lines bisecting the angles between the original axes. 15. Find the center, length of axes and the eccentricity of the ellipse 2x3y4x12y130
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ax bx dx 7. Apply differentiation under integral sign to evaluate e Join 8. Using Gamma function show that (sin x cos x dx 34 x 192 9. Find the area bounded by the curve x 4y and the line x4y2 OR Find the volume of the solid generated by the revolution of the cardioid ra (1cose) about the initial line. dy dx 10. Solve Sinx y cosxxsin x
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1. State Leibnitys theorem on Leigher derivatives If y sin (m sin x) then show that (1x) Ya2 (2n1) xyn1 (mn)yn 0 2. Assuming the validity of expansion, find the expansion of the function ex by Maclaurins 1e theorem. 3. Evaluate lim xe (1x)log(1x) x 4. Find the asymptotes of the curve y 2xyxyy10 5. Find the radius of curvature of the curve yx(x3)
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II2 dx 7. Apply differentiation under integral sign to evaluate (a sin xb cos x) dx 8. Use gamma function to prove that II3 (1x)16 9. Find the volume or surface area of solid generated by revolving the cycloid x a(0sin), y a(1cose) about its base.
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10. If the line lxmyn0 is normal to the ellipse x y 1 then show that b a b b 12 m (a2. 2 ab) n 11. Solve the locus of a point which moves in such a way that the difference of its distance from two fixed points is constant is Hyperbola. 12. Solve the differential equation x dy dy 2y6x dx dx 13. Solve (xDxD1)ysin(logx)
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5 7. Use Beta and Gamma function to evaluate (2 2axx dx sin bx 8. Evaluate dx by using the rule of differentiation under the sign of integration. 9. Find the volume of the solid formed by the revolution of the cardiod r a (1cos) about initial line. OR Find the area bounded by the curve xy a (ay) and the xaxies 40. Solve the differential equation
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12. Find the centre of the conic 3x 8xy3y 40x20y 50 0.1 13. Solve the differential equation (x y 1) 1. dx 14. Find the general solution of the differential equation p 4xyp 3y 0. 15. Find the general solution of the differential equation (D 2D 1)y e cos x. 16. Newtons law of cooling states that The temperature of an object changes at a
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1. If ya cos (logx) b sin (logx). Prove that x. yn2 (2n1)x. yn1 (n 1)yn 0. 2. State and prove Rolles theorem. 3. Determine the values of a, b, c, so that Lt (abcos x)x c. sin x 1. 0x 5 X 4. Find the asymptotes of the curve (x y) (x 2y 2) x9y2. 5. If e and e be the radii of curvature at the ends of a focal chord of the parabola y 4ax,
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6. Integrate any three X.ex .a) .dx (1x) b) f c) .dx 1ex 4 log(1 x) 2 1x .dx 2 cot x .dx 1cotx 7. Evaluate fxdx by the method of summation. 3. Obtain reduction formula for cot x dx and hence integrate fcotx dx. OR Using Gamma function show that Jex.xdxx Je fex.dx 0 9. Find the area bounded by the cardioid r a(1 cose) OR C 82 Find the volume
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1. Ify (sin x) then show that i) (1x)y2xy20 ii) (1x)y2(2n1)xyny 0 x(x3)x3,0 2. State Rolles Theorem and verify the theorem for f(x) x(x3) 3. Evaluate x 0 tan x 4. Find the asymptotes of the curve (x)(bx)x.y 5. Find the pedal equation of the curve r a cos 20 x4 6. Evaluate 4 (sin xcosx) Jo (916sn2x) dx a 7. Use Beta Gamma function to evaluate
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1. Find the angle between the curves r a sin 20, r a cos 20. OR If y (x1), prove that (x 1)yn 2 2xy 1n(n 1)yn 0. 2. State and prove Lagranges mean value theorem. 3. Evaluate lim (cot x) lo logx 0x 4. Find the surface of the right circular cylinder of greatest surface which can be inscribed in a sphere of radius r.
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7. Apply the method of differentiation under integral sign to prove x2 dx (a sin xb cos x) (a b) 4a3b3 i 8. Use Gamma function to prove that ( dx 16 o (1x) 3 9. Find the area bounded by the curve xy a(ay) and the x axis. OR Find the volume of the solid formed by revolving the cycloid x ( sine), ya(1cose) about its base. 10. Solve the
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10. Solve any three of the following differential equations. a) (3y7x7)dx (7y3x3)dy 0 b) cos x dy y(sin x y)dx c) p pyx0 where p d) (D 3D2)y xx dy dx 11. Find the changed form of the equation 3x 3y 2xy 2 when the axes are turned through 45 the origin remaining fixed. 12. The line xy0 is a directrix of an ellipse, the point (2,2) is the
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12. solve (D22D 5)ye.sinx 13. Solve the differential equation x. dx dy 4x dy 2ye dx 14. What does the equation 3x 3y 2xy 2 becomes when the axes are turned through an angle 45 to the original axis. OR 10 cos eco Describe and Sketch the graph of the conic 2 cos ec0 3 15. Derive the equation of Ellipse in the standard form. 16. Find the equation
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a tan x 1. Ify ela prove that (1x)yn2(2nx 2x a)yn1n(n 1)yn 0. 2. State and prove Lagranges mean value theorem. lim sin xx 3. Evaluate 0x X 4. Find the asymptotes of the curve (x y)(x 2y z) x9y2. 5. Find the radius of curvature of the curve ra(1 cose).
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6. Apply the method of differentiation under integral sign to evaluate o 1 tan (ax) dx. x(1x) sin 7. Prove that Jo xdx sin x cos x log(21). 2 5 8 Use Gamma function to prove cos 4 30. sin 2 60 .5 192 9. Find, by method of integration, the area of the loop of the curve ay x (ax). 10. Solve the differential equation (1x2) dy yetanx5 dx
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11. Solve yyp 2px, where p dydx 5 12. Solve (D23D2)y xx 5 13. Newtons law of cooling states that the temperature of an object changes at a rate proportional to the difference of temperature between the object and its surroundings. Supposing water at 100C cools to 80C in 10 minutes, in a room temperature of 30C, find when the temperature of
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Find the angle of intersection of the pair of curves r a cos no and r a sin no. OR If y a cos(log x) b sin (log x). Prove that xyn 2 (2n1)x. yn 1 (x 1)yn 0 StateRolles theorem and verify it for the function f(x) x.(x3).e(x2), x (3, 0 at (1x)e Evaluate x 0 X A cone is circumscribed to a sphere of radius r. Show that when the volume of the
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5. Find the asymptotes of the curve (xy) (x 2y 1) xy10. OR Show that the radius of curvature for the curve r a cos me is 6. Integrate any three m a (m1)r m1 cos xdx 4 sin 20d0 a) 4 (1 sin x)(2 sin x) sin cos c) S Vcot xdx d) 1cotx L 2 dx 7. Evaluate xdx by the method of summation.
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8. Obtain a reduction formula for sec x dx and hence find sec xdx. dx Evaluate (16) OR 9. Find the area of a loop of the curve ay ax x. OR 2 2 2 Find the volume of the solid generated by revolving the astroid x axis of x. y a3 about the
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10. Solve any three of the following differential equations. a) x dyy dx x y.dx dy b) x y.log y xy.ex dx c) y 2pxap2y0 d) (D2 3D2)y e 2 1.1. If the axes be turned through an angle tan0 2. What does the equation 4xy 3x a becomes 12. Find the equation of an ellipse in the standard form. 13. If e, and e, are the eccentricities of the
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14. Solve, yyp2px where p dx dy dy 15. Solve 32ye sin x dx dx 16. Describe and sketch the graph of the equation r 10 23sin0 OR Show that the conic section represented by the equation 14x4xy11y244x58y 710 is an ellipse. Also find its center, eccentricity, latus rectuns and foci
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If Y Sin(m sinx), then show that (1x)yn2 (2n1)xy n1 (mn)y 0 2. Apply Maclaurins series to find the expansion of lim Tan 2a (24) 3. Evaluate xa 2 ex as far as the term in x 1e 4. Find the asymptotes of the curve x(xy)23(xy)8y0 2 2 2 5. Find the pedal equation of the curve x y a 3 6. Apply the method of differentiation under integral sign to
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7. Show that log(1x) dx log2 2 1x dx 8. Use Gamma function to prove that 0 (1x)6 3 22 9. Find the area of two loops of the curve ay axx4 OR Find the volume of the solid formed by the revolution of the cycloid xa (0sin0), ya (1cos 0) about the tangent at the vertex.
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10. Solve the differential equation (1 y)(xetany) 11. Solve y3px ayp 0 12. Solve (D22D5)y ex.sin x dy 0 dx 13. A resistance of 100 Ohms, an inductance of 0.5 Henry are connected in series with a battery 20 volts. Find the current in the circuit as a function of time. 14. What does the equation 3x 3y2xy 2 becomes when the axes are turned
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1. If y log (x(a x) show that (ax)yn2 (2n1)xyn1 ny 0 2. State and prove Lagranges Mean Value theorem. lim a sin xsin 2x 3. If x 0. is finite, find the value of a and the limit. tan x 4. Find asymptotes of (xy) 2(xy) x10 (5. Find the radius of curvature at any point (x,y) for the curve x23 y23a sin bx 6. Prove that dx (b0) X 2 23
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10. Solve the differential equation dx dy y tan X 11. Solve the differential equation x dy ylogy xye dx 12. Solve the differential equation dy dy 2e e dx dx 13. Solve y px m p where p dy dx OR A resistance of 100 ohms, an inductance of 0.5 henry are connected in series with a battery of 20 volts. Find the current in the circuit as a function
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14. Solve that locus of a point which moves in such a way that the differences of it distance from two fixed point is constant is Hyperbola. 15. Find the equation of ellipse of the form x2 y2 1 where ab b 4 seco 16. Describe and sketch the graph of the equation r 2 sec01
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1. If y log(xva x), show that (a x)yn 2 (2n1)xy1ny 0. 2. State and prove Lagranges mean value theorem. lim tan x 3. Evaluate 0
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dx Use Gamma function to prove that (1x)1 16 8. Using method of differentiation under integral sign, evaluate sin bx dx. 9. Find the area bounded by the cardioid, ra(1 cos). OR Find the volume of the solid formed by revolving the cycloid x a(0 sine), ya(1 cose) about its base. Find the angle through which the axes must be turned so that the
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8. Define Beta function. Apply Beta and Gamma function to evaluate (x 2axxdx 9. Find the area common to the circle r a and the cordioid r a(1cose) 10. Through what angle should the axes be rotated to reduce the equation 3x2xy 3y2x0 into one with the xy term missing Also obtain the transformed equation. 11. Derive the equation of an ellipse in
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1. State Leibnitzs theorem on heigher order derivative. If yealanx, (1x2)2(2nx2xa)yn1n(n1)y 0 prove that 2. State difference between Rolls Theorem and Lagranges Mean value theorem. Verify Lagranges mean value theorem for f(x) x(x1)(x2) when xe 3. Define inderminate form of a function. Evaluate lim tanx x 0 4. Define asymptote to a curve. Find the
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7. Apply differentiation under integral sign to evaluate ax sin x dx. X 8. Define Beta and Gamma function. Use them to evaluate x2axxdx. 9. Show that the area of the curve x y a is OR 8 Find the volume of the solid formed by the revolution of the cardoid r a(1 cose) about the initial line. 10. Solve (1 y)dx(tanyx)dy 11. Solve y px m p where p
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12. Solve (D22D1)yex. dy 13. Solve Solve x2. d2y dx 2x 4yx. OR dx A resistance of 100 ohms, an inductance of 0.5 Henry are connected in series with a battery of 20 volts. Find the current in the circuit as a function of time. 14. What does the equation of lines 7x24xy4y2 0 become when the axes are the bisectors of the angles between them 15.
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1 1. State Leibnitzs theorem. If y (sin x), show that (1x)y2(2n1)xyny 0 2. Verify Rolles Theorem for f(x)log xab x a, b. How does Rolles Theorem (ab)x differ from Lagranges mean value theorem. lim 3. Evaluate x 0 sin x 4. Find the asymptotes to the curve y 2xyxyy10
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5. Find the radius of curvature at origin for the curve x y 3axy. 6. Show that x log(sin x)dx log 2 1 7. Apply the rule of differentiation under integral sign to evaluate sin x deduce that dx 2 ax e sin x X 0 2a dx and hence 8. Define Beta function. Apply Beta and Gamma function to evaluate (x 2axx dx 9. Find the volume generated by
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11. Find center, length of axes, eccentricity and directrices of the conic 3x8xy3y240x20y500 OR 12 Describe and sketch the conic r 26cos0 12. Deduce standard equation of ellipse. 13. Solve the differential equation (1 y)(xetan 14. Solve xp22yp ax 0 where p 15. Solve dy dy 3 2yex.sinx dx dx dy dx dy 0 dx
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PR2 absinx dx 7. Apply differentiation under integral sign to evaluate log absinx sinx 8. Define Gamma function. Apply Beta and Gamma function to evaluate cos 60.sin 30 7 192 9. Find the area inclosed by y (ax)x and its asymptotes. 10. If the axes be turned through and angle of tan 2, what does the equation 4xy3xa20 become 11. Find the center,
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12. Deduce standard equation of hyperbola. dy dx 13. Solve the differential equation x logxy2logx dy 14. Solve (xa)p (xy)py0 where p dx 15. Solve (DD2)y e sin 2x 16. Find a current i(t) in the RLC circuit assuming zero initial current and charge q, if R 80 ohms, L 20 Henry, C0.01 Fardays and E 100 volts.
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1. State Leibnitzs theorem. If y (x1), then prove that (x1)y22xy n(n1)y, 0 2. Assuming the validity of expansion, expand log(1 sin x) by Maclaurins therom. lim (1x)e 3. Evaluate x 0 X 4. Find the asymptotes of the curve x(xy)23(xy)8y0 5. Find the radius of curvature at any point (r,0) for the curve a r cos 20 6. Show that xsin x 1cos x dx 4
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Five equally qualified students (Sarah, Amy, Rahman, Firdaus and Shuan) have applied to serve as student representatives on the campus housing committee. Sarah and Shuan live in the residence hall and the other students live offcampus. A random sample of three students is selected from the five applicants to serve in the committee. Suppose a

You are a manager at Ventus LLC , a producer of onshore wind farms. You are planning two large projects, one in Massachusetts and one in New York. While some companies have already committed to purchasing energy from the wind farms, there is some risk that the projects will fail and generate a loss for the company. The CEO has asked you to give

There are 45,578 Year 12 students in Queensland that are completing QCS for the first time. The first sample of 2000 students (randomised) was collected to calculate the average test scores. From this sample the mean score was 71.9 and the standard deviation was 24.2. A 95 confidence interval was calculated which suggested the mean QCS score was

Stats Air operates a daily flight between Sydney and Griffith that has operated for six years. Flight XZ 966 is scheduled to depart Sydney at 1135 and arrive in Griffith at 1310. Flight XZ 967 is scheduled to depart Griffith at 1400 and arrive in Sydney at 1535. Both flights XZ 966 and XZ 967 are scheduled to take 95 minutes. Of course, often the

Consider the following function over the given interval. f(x) 16x2, 4,4 b Evaluate the integral, f(x) dx. 256 3 Find the average value of the function over the given interval. (Round your answer to four decimal places.) Find all values of x in the interval for which the function equals its average value. (Enter your answers as a commaseparated

ctuaries can use histograms to analyze the frequency of submitted insurance claims after mers over a particular period of time. The information in the table on the left below lists the number of flood insurance claims submitted and average paid flood claim amounts as a result of storms in 2005. An average paid claim is the average of all the flood

During the fourth wave of the pandemic 20 of the patients admitted would enter the ICU. From a sample of 100 patients admitted, what is the probability between 19 and 21 were admitted to the ICU 0.9999 0.9876 0.7887 0.1974

Suppose K numbers are chosen from {1...,n}, where k

Find the following probabilities. It may help to sketch the pdf for Standard Normal Distribution and shade in the area that corresponds to the probability of interest. 1) Z Norm(0, 1), find Pr(Z 1.78) pt 2) Z Norm(0, 1), find Pr(Z 1.96) pt 3) Z Norm(0, 1), find Pr(2.34

Two regular sixsided dice are tossed. Compute the probability that the sum of the pips on the upward faces of the two dice is the following. At most 4

The following confidence interval is obtained for a population proportion p 0.880 p 0.910 What is the margin error E.

38 If two persons are selected at random from 5 pairs of brothers and sisters, what is the probability that (i) they are brothers and sisters (ii) among them one is a boy and another is a girl
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3. Six people play a game where each player is supposed to have the same probability of winning as the others. However, Player 1 wins three times in a row. Using a 95 confidence level, should the other players have a reason to think that Player 1 is cheating

Genetic engineering for cancer treatment. Heres a new idea for treating advanced melanoma, the most serious kind of skin cancer genetically engineer white blood cells to better recognize and destroy cancer cells, then infuse these cells into patients. The subjects in a small initial study were 11 patients whose melanoma had not responded to

Suppose there is a test for checking the presence of skin cancer. When cancer is present, the test is positive 98.5 of the time and negative the other 1.5. When cancer is not present, the test is positive 3 of the time, and negative the other 97. Furthermore, the probability of having cancer is 0.8. If someone receives the test and the result is

A manager believes that the shelf life of apple juice is normally distributed. A sample of 30 containers of juice was taken and the shelf life was recorded. You are given the results below. The average shelf life in the sample was 23.07 days with a standard deviation of 4.29 days. 15 17 19 20 20 20 21 21 21 21 21 21 21 22 22 22 22 22 22 22 24 24

In a population of 10,000, there are 5,000 nonsmokers, 2,500 smokers of one pack or less per day, and 2,500 smokers of more than one pack per day. During any month, there is an 8 probability that a nonsmoker will begin smoking a pack or less per day, and a 4 probability that a nonsmoker will begin smoking more than a pack per day. For smokers who

1 The binomial distribution N B(M) (NM)M describes the probability of placing M identical particles into N equivalent physical sites, with a limit of one particlesite if the probability of a specific site being occupied is p. Using Stirlings formula (in N N in NN), show that for N M, so that p 1, the binomial distribution becomes the Poisson

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