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Probability And Statistical Inference 9th Edition Robert V. Hogg, Elliot Tanis, Dale Zimmerman - Solutions

It is claimed that 15% of the ducks in a particular region have patent schistosome infection. Suppose that seven ducks are selected at random. Let X equal the number of ducks that are infected. (a) Assuming independence, how is X distributed? (b) Find (i) P(X ≥ 2), (ii) P(X = 1), and (iii) P(X
It is believed that approximately 75% of American youth now have insurance due to the health care law. Suppose this is true, and let X equal the number of American youth in a random sample of n = 15 with private health insurance. (a) How is X distributed? (b) Find the probability that X is at least
In 2012, Red Rose tea randomly began placing 1 of 12 English porcelain miniature figurines in a l00-bag box of the tea, selecting from 12 nautical figurines. (a) On the average, how many boxes of tea must be purchased by a customer to obtain a complete collection consisting of the 12 nautical
Show that 63/512 is the probability that the fifth head is observed on the tenth independent flip of a fair coin.
Suppose an airport metal detector catches a person with metal 99% of the time. That is, it misses detecting a person with metal 1% of the time. Assume independence of people carrying metal. What is the probability that the first metal-carrying person missed (not detected) is among the first 50
Use the result of Exercise 2.5-5 to find the mean and variance of the (a) Bernoulli distribution. (b) Binomial distribution. (c) Geometric distribution. (d) Negative binomial distribution.
The mean of a Poisson random variable X is μ = 9. Compute P(μ − 2σ < X < μ+ 2σ).
Find P(X = 4) if X has a Poisson distribution such that 3P(X = 1) = P(X = 2).
Suppose that the probability of suffering a side effect from a certain flu vaccine is 0.005. If 1000 persons are inoculated, find the approximate probability that At most 1 person suffers. 4, 5, or 6 persons suffer
The pdf of X is f(x) = c/x2, 1 < x < ∞.(a) Calculate the value of c so that f(x) is a pdf.(b) Show that E(X) is not finite
Sketch the graphs of the following pdfs and find and sketch the graphs of the cdfs associated with these distributions (note carefully the relationship between the shape of the graph of the pdf and the concavity of the graph of the cdf):
Let f(x) = 1/2, 0 < x < 1 or 2 < x < 3, zero elsewhere, be the pdf of X. (a) Define the cdf of X and sketch its graph. (b) Find q1 = π0.25. (c) Find m = π0.50. Is it unique? (d) Find q3 = π0.75.
Let f(x) = (x + 1)/2, −1 < x < 1. Find (a) π0.64, (b) q1 = π0.25, and (c) π0.81.
The weekly demand X for propane gas (in thousands of gallons) has the pdfIf the stockpile consists of two thousand gallons at the beginning of each week (and nothing extra is received during the week), what is the probability of not being able to meet the demand during a given week?
Let f(x) = 1/2, −1 ≤ x ≤ 1, be the pdf of X. Graph the pdf and cdf, and record the mean and variance of X.
Nicol lets the pdf of X be defined byFind(a) The value of c so that f(x) is a pdf.(b) The mean of X (if it exists).(c) The variance of X (if it exists).(d) P(1/2 ‰¤ X ‰¤ 2).
If the mgf of X isFind (a) E(X), (b) Var(X), and (c) P(4.2
A grocery store has n watermelons to sell and makes $1.00 on each sale. Say the number of consumers of these watermelons is a random variable with a distribution that can be approximated byA pdf of the continuous type. If the grocer does not have enough watermelons to sell to all consumers, she
For each of the following functions,(i) Find the constant c so that f(x) is a pdf of a random variable X,(ii) Find the cdf, F(x) = P(X ‰¤ x),(iii) Sketch graphs of the pdf f(x) and the distribution function F(x), and(iv) Find Î¼ and Ïƒ 2:
Use the moment-generating function of a gamma distribution to show that E(X) = αθ and Var(X) = αθ2
Let X equal the number of alpha particle emissions of carbon-14 that are counted by a Geiger counter each second. Assume that the distribution of X is Poisson with mean 16. Let W equal the time in seconds before the seventh count is made. (a) Give the distribution of W. (b) Find P(W ≤ 0.5).
Cars arrive at a tollbooth at a mean rate of 5 cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.30 minutes before collecting the eighth toll.
Say the serum cholesterol level (X) of U.S. males ages 25€“34 follows a translated gamma distribution with pdf(a) What are the mean and the variance of this distribution?(b) What is the mode?(c) What percentage have a serum cholesterol level less than 200?
Telephone calls arrive at a doctor’s office according to a Poisson process on the average of two every 3 minutes. Let X denote the waiting time until the first call that arrives after 10 a.m. (a) What is the pdf of X? (b) Find P(X > 2).
The initial value of an appliance is $700 and its dollar value in the future is given byWhere t is time in years. Thus, after the first three years, the appliance is worth nothing as far as the warranty is concerned. If it fails in the first three years, the warranty pays v(t). Compute the expected
Let X have a logistic distribution with pdfShow that Has a U(0, 1) distribution.
Let the random variable X be equal to the number of days that it takes a high-risk driver to have an accident. Assume that X has an exponential distribution. If P(X < 50) = 0.25, compute P(X > 100 | X > 50).
Let F(x) be the cdf of the continuous-type random variable X, and assume that F(x) = 0 for x ≤ 0 and 0 < F(x) < 1 for 0 < x. Prove that if P(X > x + y | X > x) = P(X > y), Then F(x) = 1 − e−λx, 0 < x. Which implies that g(x) = acx.
A certain type of aluminum screen 2 feet in width has, on the average, three flaws in a l00-foot roll. (a) What is the probability that the first 40 feet in a roll contain no flaws? (b) What assumption did you make to solve part (a)?
If X has a gamma distribution with θ = 4 and α = 2, find P(X < 5).
If X is N(μ, σ2), show that the distribution of Y = aX + b is N(aμ + b, a2σ2), a ≠ 0.
If the moment-generating function of X is given by M(t) = e500t+5000t2, find P[27, 060 ≤ (X − 500)2 ≤ 50, 240].
The strength X of a certain material is such that its distribution is found by X = eY, where Y is N(10, 1). Find the cdf and pdf of X, and compute P(10, 000 < X < 20, 000).
The graphs of the moment-generating functions of three normal distributions€”N(0, 1), N(ˆ’1, 1), and N(2, 1)€”are given in Figure 3.3-3(a). Identify them.
If Z is N(0, 1), find (a) P(0 ≤ Z ≤ 0.87). (b) P(−2.64 ≤ Z ≤ 0). (c) P(−2.13 ≤ Z ≤ −0.56). (d) P(|Z| > 1.39). (e) P(Z < −1.62). (f) P(|Z| > 1). (g) P(|Z| > 2). (h) P(|Z| > 3).
Find the values of (a) z0.10, (b) −z0.05, (c) −z0.0485, and (d) z0.9656.
If the moment-generating function of X is M(t) = exp(166t + 200t2), find (a) The mean of X. (b) The variance of X. (c) P(170 < X < 200). (d) P(148 ≤ X ≤ 172).
Let the distribution of X be N(μ, σ2). Show that the points of inflection of the graph of the pdf of X occur at x = μ ± σ.
The weekly gravel demand X (in tons) follows the PdfHowever, the owner of the gravel pit can produce at most only 4 tons of gravel per week. Compute the expected value of the tons sold per week by the owner.
Let X have an exponential distribution with θ = 1; that is, the pdf of X is f(x) = e−x, 0 < x < ∞. Let T be defined by T = ln X, so that the cdf of T is G(t) = P(ln X ≤ t) = P(X ≤ et) (a) Show that the pdf of T is g(t) = ete−et, −∞ < x < ∞, which is the pdf of an extreme-value
A customer buys a $1000 deductible policy on her $31,000 car. The probability of having an accident in which the loss is greater than $1000 is 0.03, and then that loss, as a fraction of the value of the car minus the deductible, has the pdf f(x) = 6(1 − x)5, 0 < x < 1. (a) What is the probability
A certain machine has a life X that has an exponential distribution with mean 10. The warranty is such that $m is returned if the machine fails in the first year, (0.5)m of the price is returned for a failure during the second year, and nothing is returned after that. If the machine cost $2500,
The time X to failure of a machine has pdfCompute P(X > 5 | X > 4).
Suppose that the length W of a man’s life does follow the Gompertz distribution with λ(w) = a(1.1)w = ae(ln 1.1)w, P(63 < W < 64) = 0.01. Determine the constant a and P(W ≤ 71 | 70 < W).
Let X be the failure time (in months) of a certain insulating material. The distribution of X is modeled by the pdfFind (a) P(40 (b) P(X > 80)
A frequent force of mortality used in actuarial science is λ(w) = aebw + c. Find the cdf and pdf associated with this Makeham’s law.
Determine the indicated probabilities from the graph of the second cdf of X in Figure 3.4-4:(a) P (ˆ’1/2 ‰¤ X ‰¤ ½) . (b) P (1/2 (c) P (3/4 (d) P(X > 1). (e) P(2 (f) P(2
Find the mean and variance of X if the cdf of X is
Roll a pair of four-sided dice, one red and one black, each of which has possible outcomes 1, 2, 3, 4 that have equal probabilities. Let X equal the outcome on the red die, and let Y equal the outcome on the black die. (a) On graph paper, show the space of X and Y. (b) Define the joint pmf on the
Select an (even) integer randomly from the set {0, 2, 4, 6, 8}. Then select an integer randomly from the set {0, 1, 2, 3, 4}. Let X equal the integer that is selected from the first set and let Y equal the sum of the two integers. (a) Show the joint pmf of X and Y on the space of X and Y. (b)
The torque required to remove bolts in a steel plate is rated as very high, high, average, and low, and these occur about 30%, 40%, 20%, and 10% of the time, respectively. Suppose n = 25 bolts are rated; what is the probability of rating 7 very high, 8 high, 6 average, and 4 low? Assume
In a smoking survey among boys between the ages of 12 and 17, 78% prefer to date nonsmokers, 1% prefer to date smokers, and 21% don’t care. Suppose seven such boys are selected randomly. Let X equal the number who prefer to date nonsmokers and Y equal the number who prefer to date smokers. (a)
If the correlation coefficient ρ exists, show that ρ satisfies the inequality −1 ≤ ρ ≤ 1.
Let X and Y have the joint pmf defined by f(0, 0) = f(1, 2) = 0.2, f(0, 1) = f(1, 1) = 0.3. (a) Compute μX, μY, σ2X, σ2Y, Cov(X, Y), and ρ. (b) Find the equation of the least squares regression line and draw it on your graph. Does the line make sense to you intuitively?
Let X and Y have a trinomial distribution with parameters n = 3, pX = 1/6, and pY = 1/2. Find (a) E(X). (b) E(Y). (c) Var(X). (d) Var(Y). (e) Cov(X, Y). (f) ρ.
The joint pmf of X and Y is f(x, y) = 1/6, 0 ≤ x + y ≤ 2, where x and y are nonnegative integers. (a) Sketch the support of X and Y. (b) Record the marginal pmfs fX(x) and fY(y) in the “margins.” (c) Compute Cov(X, Y). (d) Determine ρ, the correlation coefficient. (e) Find the best-fitting
A certain raw material is classified as to moisture content X (in percent) and impurity Y (in percent). Let X and Y have the joint pmf given by(a) Find the marginal pmfs, the means, and the variances. (b) Find the covariance and the correlation coefficient of X and Y. (c) If additional heating is
Let fX(x) = 1/10, x = 0, 1, 2, . . . , 9, and h(y | x) = 1/(10 − x), y = x, x + 1, . . . , 9. Find (a) f(x, y). (b) fY(y). (c) E(Y | x).
Let the joint pmf f(x, y) of X and Y be given by the following: (x, y) f(x, y) (1, 1) ....... 3/8 (2, 1) ....... 1/8 (1, 2) ....... 1/8 (2, 2) ....... 3/8 Find the two conditional probability mass functions and the corresponding means and variances.
The alleles for eye color in a certain male fruit fly are (R, W). The alleles for eye color in the mating female fruit fly are (R, W). Their offspring receive one allele for eye color from each parent. If an offspring ends up with either (W, W), (R, W), or (W, R), its eyes will look white. Let X
An insurance company sells both homeowners€™ insurance and automobile deductible insurance. Let X be the deductible on the homeowners€™ insurance and Y the deductible on automobile insurance. Among those who take both types of insurance with this company, we find the following
A fair six-sided die is rolled 30 independent times. Let X be the number of ones and Y the number of twos. (a) What is the joint pmf of X and Y? (b) Find the conditional pmf of X, given Y = y. (c) Compute E(X2 − 4XY + 3Y2).
Let T1 and T2 be random times for a company to complete two steps in a certain process. Say T1 and T2 are measured in days and they have the joint pdf that is uniform over the space 1 < T1 < 10, 2 < T2 < 6, T1 + 2T2 < 14. What is P(T1 + T2 > 10)?
Show that in the bivariate situation, E is a linear or distributive operator. That is, for constants a1 and a2, show that
Let X and Y be random variables of the continuous type having the joint pdf f(x, y) = 8xy, 0 ≤ x ≤ y ≤ 1. Draw a graph that illustrates the domain of this pdf. (a) Find the marginal pdfs of X and Y. (b) Compute μX, μY, σ2X, σ2Y, Cov(X, Y), and ρ. (c) Determine the equation of the least
For the random variables defined in Example 4.4-3, calculate the correlation coefficient directly from theDefinition
Let f (x, y) = 1/8, 0 ≤ y ≤ 4, y ≤ x ≤ y + 2, be the joint pdf of X and Y.(a) Sketch the region for which f (x, y) > 0.(b) Find fX(x), the marginal pdf of X.(c) Find fY(y), the marginal pdf of Y.(d) Determine h(y | x), the
Let X and Y have the joint pdf f(x, y) = x + y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (a) Find the marginal pdfs fX(x) and fY(y) and show that f(x, y) ≠ fX(x)fY(y). Thus, X and Y are dependent. (b) Compute (i) μX, (ii) μY, (iii) σ2X, and (iv) σ2Y.
Let X have a uniform distribution on the interval (0,1). Given that X = x, let Y have a uniform distribution on the interval (0, x + 1). (a) Find the joint pdf of X and Y. Sketch the region where f(x, y) > 0. (b) Find E(Y | x), the conditional mean of Y, given that X = x. Draw this line on the
Let f(x, y) = 3/2, x2 ≤ y ≤ 1, 0 ≤ x ≤ 1, be the joint pdf of X and Y. (a) Find P(0 ≤ X ≤ 1/2). (b) Find P(1/2 ≤ Y ≤ 1). (c) Find P(X ≥ 1/2, Y ≥ 1/2). (d) Are X and Y independent? Why or why not?
Using Example 4.4-2, (a) Determine the variances of X and Y. (b) Find P(−X ≤ Y).
Using the background of Example 4.4-4, calculate the means and variances of X and Y.
In a college health fitness program, let X denote the weight in kilograms of a male freshman at the beginning of the program and Y denote his weight change during a semester. Assume that X and Y have a bivariate normal distribution with μX = 72.30, σ2x = 110.25, μY = 2.80, σ2y = 2.89, and ρ =
LetShow that f(x, y) is a joint pdf and the two marginal pdfs are each normal. Note that X and Y can each be normal, but their joint pdf is not bivariate normal.
Show that the expression in the exponent of Equation 4.5-2 is equal to the function q(x, y) given in the text.
Let X and Y have a bivariate normal distribution with μX = 70, σ2X = 100, μY = 80, σ2Y = 169, and ρ = 5/13. Find (a) E(Y | X = 72). (b) Var(Y| X = 72). (c) P(Y ≤ 84, | X = 72).
For a freshman taking introductory statistics and majoring in psychology, let X equal the student’s ACT mathematics score and Y the student’s ACT verbal score. Assume that X and Y have a bivariate normal distribution with μX = 22.7, σ2x = 17.64, μY = 22.7, σ2Y = 12.25, and ρ = 0.78. (a)
Let X and Y have a bivariate normal distribution with parameters μX = 10, σ2x = 9, μY = 15, σY2 = 16, and ρ = 0. Find (a) P(13.6 < Y < 17.2). (b) E(Y | x). (c) Var(Y | x). (d) P(13.6 < Y < 17.2 | X = 9.1).
Let X have the uniform distribution U(−1, 3). Find the pdf of Y = X2.
Let f(x) = 1/[π(1 + x2)], −∞ < x < ∞, be the pdf of the Cauchy random variable X. Show that E(X) does not exist.
Let X be N(0, 1). Find the pdf of Y = |X|, a distribution that is often called the half-normal. Hint: Here y ∈ S y = {y : 0 < y < ∞}. Consider the two transformations x1 = −y, −∞ < x1 < 0, and x2 = y, 0 < y < ∞.
Let X have theFind the pdf of Y = X2.
The pdf of X is f(x) = 2x, 0 < x < 1. (a) Find the cdf of X. (b) Describe how an observation of X can be simulated. (c) Simulate 10 observations of X.
Let X have a logistic distribution with pdfShow that Has a U(0, 1) distribution.
The lifetime (in years) of a manufactured product is Y = 5X0.7, where X has an exponential distribution with mean 1. Find the cdf and pdf of Y.
When Î± and Î² are integers and 0Where n = Î± + Î² ˆ’ 1. Verify this formula when Î± = 4 and Î² = 3.
Let W1, W2 be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, (W1 + W2)/2.(a) Show that the pdf of X1 = (1/2)W1 is(b) Let Y1 = X1 + X2 = W and Y2 = X1, where X2 = (1/2)W2. Show that the joint pdf of Y1 and Y2 is(c) Show that the pdf of Y1 = W is
A company provides earthquake insurance. The premium X is modeled by the pdfWhile the claims Y have the pdf If X and Y are independent, find the pdf of Z = X/Y.
Let W have an F distribution with parameters r1 and r2. Show that Z = 1/[1 + (r1/r2)W] has a beta distribution.
Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1 = (X1/r1)/(X2/r2) and Y2 = X2. (a) Find the joint pdf of Y1 and Y2. (b) Determine the marginal pdf of Y1 and show that Y1 has an F distribution.
Let the distribution of W be F(9, 24). Find the following: (a) F0.05(9, 24). (b) F0.95(9, 24). (c) P(0.277 ≤ W ≤ 2.70).
Let X1 and X2 have independent gamma distributions with parameters Î±, Î¸ and Î², Î¸, respectively. Let W = X1/(X1 + X2). Use a method similar to that given in the derivation of the F distribution (Example 5.2-4) to show that the pdf of W isWe say
Let X have a beta distribution with parameters Î± and Î².(a) Show that the mean and variance of X are, respectively(b) Show that when Î± > 1 and Î² > 1, the mode is at x = (Î± ˆ’ 1)/(Î± + Î² ˆ’
Let X1, X2, X3 denote a random sample of size n = 3 from a distribution with the geometric pmf(a) Compute P(X1 = 1, X2 = 3, X3 = 1). (b) Determine P(X1 + X2 + X3 = 5). (c) If Y equals the maximum of X1, X2, X3, find P(Y ‰¤ 2) = P(X1 ‰¤ 2)P(X2 ‰¤ 2)P(X3
Let X1, X2, X3 be a random sample of size n = 3 from the exponential distribution with pdf f(x) = e−x, 0 < x < ∞. Find P(1 < min Xi) = P(1 < X1, 1 < X2, 1 < X3)
Let X1, X2, X3 be independent random variables that represent lifetimes (in hours) of three key components of a device. Say their respective distributions are exponential with means 1000, 1500, and 2000. Let Y be the minimum of X1, X2, X3 and compute P(Y > 1000).
Each of eight bearings in a bearing assembly has a diameter (in millimeters) that has the pdf f(x) = 10x9, 0 < x < 1. Assuming independence, find the cdf and the pdf of the maximum diameter (say, Y) of the eight bearings and compute P(0.9999 < Y < 1).
The lifetime in months of a certain part has a gamma distribution with α = θ = 2. A company buys three such parts and uses one until it fails, replacing it with a second part. When the latter fails, it is replaced by the third part. What are the mean and the variance of the total lifetime (the
Let X1 and X2 be independent random variables with respective binomial distributions b(3, 1/2) and b(5, 1/2). Determine (a) P(X1 = 2, X2 = 4). (b) P(X1 + X2 = 7).
Let X and Y be independent random variables with nonzero variances. Find the correlation coefficient of W = XY and V = X in terms of the means and variances of X and Y.

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