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elementary probability for applications
Probability An Introduction 2nd Edition Geoffrey Grimmett, Dominic Welsh - Solutions
(i) ebtM1(at ),
(b) Let X1 and X2 be independent random variables with moment generating functions M1(t ) and M2(t ). Find random variables with the following moment generating functions:
of X, and hence find E(X3).
24. (a) Let X be an exponential random variable with parameter λ. Find the moment generating
Find the moment generating function of Xn.Let Y1, Y2, . . . , Yn be independent random variables, each having the same distribution as X1.Find the moment generating function of Pn i=1 Yi , and deduce its distribution. (Oxford 2005)
23. Define the moment generating function of a random variable X.If X and Y are independent random variables with moment generating functions MX (t ) and MY (t ), respectively, find the moment generating function of X + Y .For n = 1, 2, . . . , let Xn have probability density function fn(x) =1(n
(b) There are N horses that compete in m races. The results of different races are independent.The probability of horse i winning any given race is pi ≥ 0, with p1+p2+· · ·+pN = 1.Let Q be the probability that the same horse wins all m races. Express Q as a polynomial of degree m in the
22. (a) Suppose that f (x) = xm, where m is a positive integer, and X is a random variable taking values x1, x2, . . . , xN ≥ 0 with equal probabilities, and where the sum x1 + x2 + · · · + xN = 1. Deduce from Jensen’s inequality that XN i=1 f (xi ) ≥ N f (1/N).
where an and bn are arbitrary real numbers.Suppose that φ(t ) = e−|t |α, where 0 < α ≤ 2. Determine an and bn such that Yn has the same distribution as X1 for n = 1, 2, . . . . Find the probability density functions of X1 when α = 1 and when α = 2. (Oxford 1980F)
21. Let X1, X2, . . . , Xn be independent random variables, each with characteristic function φ(t).Obtain the characteristic function of Yn = an + bn(X1 + X2 + · · · + Xn),
Obtain the moment generating function of X and hence show that if R is another independent random variable with P(R = r ) = (e − 1)e−r for r = 1, 2, . . . , then R − X is exponentially distributed. (Oxford 1981F)
Let X = max{U1,U2, . . . ,UN }, where the Ui are independent random variables uniformly distributed on (0, 1) and N is an independent random variable whose distribution is given by P(N = k) =1(e − 1)k!for k = 1, 2, . . . .
20. Let X be a random variable whose moment generating function M(t ) exists for |t | < h, where h > 0. Let N be a random variable taking positive integer values such that P(N = k) > 0 for k = 1, 2, . . . .Show that M(t ) =∞X k=1 P(N = k)E(et X | N = k) for |t | < h.
19. Show that φ(t ) = exp(−|t |α) can be the characteristic function of a distribution with finite variance if and only if α = 2.
Show that An has the Cauchy distribution regardless of the value of n.
18. Let X1, X2, . . . be independent random variables each having the Cauchy distribution, and let An =1 n(X1 + X2 + · · · + Xn).
17. Find the characteristic function of a random variable with density function f (x) = 1 2 e−|x| for x ∈ R.132 Moments, and moment generating functions
16. Show that X and −X have the same distribution if and only if φX is a purely real-valued function.
15. Prove that if φ1 and φ2 are characteristic functions, then so is φ = αφ1 + (1 − α)φ2 for anyα ∈ R satisfying 0 ≤ α ≤ 1.
14. Coupon-collecting problem. There are c different types of coupon, and each coupon obtained is equally likely to be any one of the c types. Find the moment generating function of the total number N of coupons which you must collect in order to obtain a complete set.
(a) Show that M(t)M(−t) is the moment generating function of X − Y, where Y is independent of X but has the same distribution.(b) In a similar way, describe random variables which have moment generating functions 12 − M(t ), Z∞0 M(ut )e−u du.
13. Let X have moment generating function M(t).
Deduce the joint moment generating function of a pair of random variables having the bivariate normal distribution (6.76) with parameters μ1,μ2, σ1, σ2, ρ.* 12. Let X and Y be independent random variables, each having mean 0, variance 1, and finite moment generating function M(t ). If X + Y
11. The joint moment generating function of two random variables X and Y is defined to be the function M(s, t ) of two real variables defined by M(s, t) = E(esX+tY )for all values of s and t for which this expectation exists. Show that the jointmoment generating function of a pair of random
The winner of a game scores 2 points, the loser none; if a game is drawn, each player scores 1 point. Let X and Y be the number of points scored by A and B, respectively, in a series of n games. Prove that cov(X, Y ) = −2npq. (Oxford 1982M)
Two players A and B play a series of independent games. The probability that A wins any particular game is p2, that B wins is q2, and that the game is a draw is 2pq, where p+q = 1.
10. Prove that if X = X1 + · · · + Xn and Y = Y1 + · · · + Yn, where Xi and Y j are independent whenever i 6= j , then cov(X, Y ) =Pn i=1 cov(Xi , Yi ). (Assume that all series involved are absolutely convergent.)
7.7 Problems 131 are uncorrelated and have unit variance. Show that this will be the case if and only if AV A′ = I, and show that A can be chosen to satisfy this equation if and only if V is non-singular. (Any standard results from matrix theory may, if clearly stated, be used without proof. A′
Calculate the variance of the random variable Z =XN n=1 an Xn, and deduce that the symmetric matrix V = (vmn ) is non-negative definite. It is desired to find an N × N matrix A such that the random variables Yn =XN r=1
9. Random variables X1, X2, . . . , XN have zero expectations, and E(Xm Xn) = vmn for m, n = 1, 2, . . . , N.
8. Let X1, X2, . . . be independent, identically distributed random variables and let N be a random variable which takes values in the positive integers and is independent of the Xi . Find the moment generating function of S = X1 + X2 + · · · + XN in terms of the moment generating functions of N
Deduce that, if X1, X2, . . . , Xn are independent random variables having the normal distribution with mean 0 and variance 1, then Z = X2 1 + X2 2 + · · · + X2 nhas the χ2 distribution with n degrees of freedom. Hence or otherwise show that the sum of two independent random variables, having
7. Show from the result of Problem 7.7.5 that the χ2 distribution with n degrees of freedom has moment generating function M(t ) = (1 − 2t )−1 2 n if t < 1 2 .
6. Let X1, X2, . . . , Xn be independent random variables with the exponential distribution, parameterλ. Show that X1 + X2 + · · · + Xn has the gamma distribution with parameters n andλ.
5. Let X and Y be independent random variables, X having the gamma distribution with parameters s and λ, and Y having the gamma distribution with parameters t and λ. Use moment generating functions to show that X +Y has the gamma distribution with parameters s +t andλ.
4. Show that every distribution function has only a countable set of points of discontinuity.
This fact is of importance in statistics and is used when estimating the population variance from knowledge of a random sample.130 Moments, and moment generating functions 3. Let X1, X2, . . . be identically distributed, independent random variables and let Sn = X1 + X2 + · · · + Xn. Show that
2. Let X1, X2, . . . be uncorrelated random variables, each having mean μ and variance σ 2. If X = n−1(X1 + X2 + · · · + Xn), show that E 1 n − 1 Xn i=1(Xi − X)2 = σ 2.
1. Let X and Y be random variables with equal variance. Show that U = X −Y and V = X +Y are uncorrelated. Give an example to show that U and V need not be independent even if, further, X and Y are independent.
since the integrand in the latter integral is the density function of the normal distribution with mean t and variance 1, and thus has integral 1. The moment generating function MX (t) exists for all t ∈ R. △Example 7.45 If X has the exponential distribution with parameter λ, then MX (t)
If X has the normal distribution with mean 0 and variance 1, then MX (t) =Z∞−∞et x 1√2πe−1 2 x2 dx= e 12 t 2 Z∞−∞1√2πe−1 2 (x−t)2 dx= e 12 t 2, (7.44)
σ 2. If Sn = X1 + X2 + · · · + Xn, show that cov(Sm, Sn) = var(Sm) = mσ 2 if m < n.Exercise 7.37 Show that cov(X, Y) = 1 in the case when X and Y have joint density function f (x, y) =1 ye−y−x/y if x, y > 0, 0 otherwise.
Exercise 7.36 Let X1, X2, . . . be a sequence of uncorrelated random variables, each having variance
Exercise 7.35 If X and Y have the bivariate normal distribution with parameters μ1, μ2, σ1, σ2, ρ (see(6.76)), show that cov(X, Y) = ρσ1σ2 and ρ(X, Y ) = ρ.
If ρ(X, Y ) = 0, we say that X and Y are uncorrelated.
(b) Y is a linear increasing function of X if and only if ρ(X, Y ) = 1,(•c) Y is a linear decreasing function of X if and only if ρ(X, Y ) = −1.
(a) if X and Y are independent, then ρ(X, Y ) = 0,
Exercise 7.12 If X has the χ2 distribution with n degrees of freedom, show that E(Xk ) = 2k Ŵ(k + 1 2 n)Ŵ( 1 2n)for k = 1, 2,
Exercise 7.11 If X has the gamma distribution with parameters w and λ, show that E(Xk ) =Ŵ(w + k)λkŴ(w)for k = 1, 2, . . . .
Exercise 7.10 If X is uniformly distributed on (a, b), show that E(Xk ) =bk+1 − ak+1(b − a)(k + 1)for k = 1, 2, . . . .
(b) f has finite moments of all orders,(c) fa and f have equal moments of all orders, in that Z∞−∞xk f (x) dx =Z∞−∞xk fa(x) dx for k = 1, 2, . . . .Thus, { fa : −1 ≤ a ≤ 1} is a collection of distinct density functions having the same moments.△
(a) fa is a density function,
(d) Given that the two roots R1, R2 of the above quadratic are real, what is the probability that both |R1| ≤ 1 and |R2| ≤ 1?(Cambridge 2012)If X has the Cauchy distribution, then E(Xk ) = Z ∞−∞xk π(1 + x2)dx
(c) What is the probability that the random quadratic equation x2 + 2V x + U = 0 has real roots?
(b) Let U, V be independent random variables, each with the uniform distribution on [0, 1].Show that P(V2 > U > x) = 1 3 − x + 2 3 x3/2 for x ∈ (0, 1).
28. (a) Define the distribution function F of a random variable, and also its density function f , assuming F is differentiable. Show that f (x) = −d dx P(X > x).
(iii) Let X and Y be independent normal random variables, each with mean 0, and with non-zero variances a2 and b2, respectively. Show that V = Y/X has probability density function fV (v) =cπ(c2 + v2)for −∞ < v < ∞, where c = b/a. Hence find P(|Y | < |X|)(•b) Now let X and Y be independent
27. (a) Suppose that the continuous random variables X and Y are independent with probability density functions f and g, both of which are symmetric about zero.(i) Find the joint probability density function of (U, V ), where U = X and V = Y/X.(ii) Show that the marginal density function of V is fV
Show that the joint probability density function of U = 1 2 (X − Y) and V = Y is fU,V (u, v) =(1 2 e−u−v if (u, v) ∈ A, 0 otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has probability density function fU(u) = 1 2 e−|u|, −∞ < u < ∞.(Oxford 2008)
Find the density of X + 1 2Y . Is it the same as the density of max{X, Y}? (Cambridge 2007)25. Let X and Y have the bivariate normal density function f (x, y) =1 2πp 1 − ρ2 exp−1 2(1 − ρ2)(x2 − 2ρxy + y2)for x, y ∈ R, for fixed ρ ∈ (−1, 1). Let Z = (Y − ρX)/p 1 − ρ2. Show
Let X and Y be independent and exponentially distributed random variables, each with density f (x) = λe−λx for x ≥ 0.
24. Let X and Y be independent non-negative random variables with densities f and g, respectively.Find the joint density function of U = X and V = X + aY, where a is a positive constant.
Deduce that the probability that all three crew members can keep in touch is (π + 2)/(4π).* 23. Zog continued. This time, n members of DrWho’s crew are transported to Zog, their positions being independent and uniformly distributed on the surface. In addition, DrWho is required to choose a
(b) Find the probability that Z is in direct radio communication with both X and Y, conditional on the event that φ > 1 2π.
* 22. Three crew members of Dr Who’s spacecraft Tardis are teleported to the surface of the spherical planet Zog. Their positions X, Y , Z are independent and uniformly distributed on the surface. Find the probability density function of the angle[XCY, where C is the centre of Zog.Two people
21. Let X and Y be independent random variables, each uniformly distributed on [0, 1]. Let U =min{U, V} and V = max{U, V}. Show that E(U) = 1 3 , and hence find the covariance of U and V . (Cambridge 2007)
Find the probability density function fX (x) of X, and find also E(X). Find the conditional probability density function fY |X (y | x) of Y given that X = x, and find also E(Y | X = x).(Oxford 2005)
20. Let X and Y be random variables with the vector (X, Y) uniformly distributed on the region R = {(x, y) : 0 < y < x < 1}. Write down the joint probability density function of (X, Y).Find P(X + Y < 1).
(b) Deduce that the following random variables are independent:X1/r1 X2/r2 and X3/r3(X1 + X2)/(r1 +r2).(Oxford 1982F)
(a) Show that Y1 = X1/X2 and Y2 = X1 + X2 are independent and that Y2 is a χ2 random variable with r1 +r2 degrees of freedom.
19. Let X1, X2, X3 be independent χ2 random variables with r1, r2, r3 degrees of freedom.
Prove that U and V are independent, and find their distributions.Deduce that EX X + Y=E(X)E(X) + E(Y).(Oxford 1971F)
18. Leta, b > 0. Independent positive random variables X and Y have probability densities 1Ŵ(a)xa−1e−x , 1Ŵ(b)yb−1e−y , for x, y ≥ 0, respectively, and U and V are defined by U = X + Y, V =X X + Y.
17. In a sequence of dependent Bernoulli trials, the conditional probability of success at the i th trial, given that all preceding trials have resulted in failure, is pi (i = 1, 2, . . . ). Give an expression in terms of the pi for the probability that the first success occurs at the nth
6.9 Problems 105
15. Let X and Y be independent random variables, X having the normal distribution with mean 0 and variance 1, and Y having the χ2 distribution with n degrees of freedom. Show that T =X√Y/n has density function f (t ) =1√πnŴ( 1 2 (n + 1))Ŵ( 1 2n)1 +t2 n!−1 2 (n+1)for t ∈ R.T is said to
(c) Find the mean of V/U, where U = max{|X|, |Y|} and V = min{|X|, |Y|}.(Oxford 1985M)
(b) Find the mean of Z = X2/(X2 + Y2).
(a) Show that W = 2X − Y is normally distributed, and find its mean and variance.
14. The independent random variables X and Y are normally distributed with mean 0 and variance 1.
(a) Find the (cumulative) distribution and density functions of the random variables 1−e−λX , min{X, Y }, and X − Y .(b) Find the probability that max{X, Y } ≤ aX, where a is a real constant.(Oxford 1982M)
(b) 0 < tan−1(Y/X) < α and Y > 0.(Consider various cases depending on the relative sizes ofa, b, and c.) (Oxford 1981M)13. The independent random variables X and Y are both exponentially distributed with parameterλ, that is, each has density function f (t ) =(λe−λt if t > 0, 0 otherwise.
12. X and Y are independent random variables normally distributed with mean zero and varianceσ 2. Find the expectation of pX2 + Y 2. Find the probabilities of the following events, wherea,b, c, and α are positive constants such that b < c and α < 1 2π:(a)p X2 + Y 2 < a,
11. An aeroplane drops medical supplies to two duellists. With respect to Cartesian coordinates whose origin is at the target point, both the x and y coordinates of the landing point of the supplies have normal distributions which are independent. These two distributions have the same mean 0 and
9. Let X and Y have joint density function f (x, y) =(1 4 (x + 3y)e−(x+y) if x, y ≥ 0, 0 otherwise.Find the marginal density function of Y. Show that P(Y > X) = 5 8 .10. Let Sn be the sum of n independent, identically distributed random variables having the exponential distribution with
8. Show that there exists a constant c such that the function f (x, y) =c(1 + x2 + y2)3/2 for x, y ∈ R is a joint density function. Show that both marginal density functions of f are the density function of the Cauchy distribution.
7. Let X1, X2, . . . be independent, identically distributed, continuous random variables. Define N as the index such that X1 ≥ X2 ≥ · · · ≥ XN−1 and XN−1 < XN .Prove that P(N = k) = (k − 1)/k! and that E(N) = e.
6. Let X1, X2, . . . , Xn be independent random variables, each having distribution function F and density function f . Find the distribution function of U and the density functions of U and V , where U = min{X1, X2, . . . , Xn} and V = max{X1, X2, . . . , Xn}. Show that the joint density function
5. Lack-of-memory property. If X has the exponential distribution, show that P????X > u + v X > u= P(X > v) for u, v > 0.This is called the ‘lack of memory’ property, since it says that, if we are given that X > u, then the distribution of X − u is the same as the original distribution of X.
4. Show that if X and Y are independent random variables having the exponential distribution with parameters λ and μ, respectively, then min{X, Y } has the exponential distribution with parameter λ + μ.
3. Let (X, Y, Z) be a point chosen uniformly at random in the unit cube (0, 1)3. Find the probability that the quadratic equation Xt 2 + Y t + Z = 0 has two distinct real roots.
1 if x + y ≥ 0, 0 otherwise, the joint distribution function of some pair of random variables? Justify your answer.6.9 Problems 103
1. If X and Y are independent random variables with density functions fX and fY , respectively, show that U = XY and V = X/Y have density functions fU (u) =Z∞−∞fX (x) fY (u/x)1|x|dx, fV (v) =Z∞−∞fX (vy) fY (y)|y| dy.2. Is the function G, defined by G(x, y) =(
Exercise 6.80 Let the pair (X, Y) have the bivariate normal distribution of (6.76), and leta, b ∈ R.Show that aX + bY has a univariate normal distribution, possibly wth zero variance.
Exercise 6.79 Let the pair (X, Y ) have the bivariate normal density function of (6.76), and let U and V be given by (6.78). Show that U and V have the standard bivariate normal distribution. Hence or otherwise show that E(XY ) − E(X)E(Y ) = ρσ1σ2, and that E(Y | X = x) = μ2 + ρσ2(x −
Show that the joint density function of U = 1 2 (X − Y ) and V = Y is fU,V (u, v) =(1 2 e−u−v if (u, v) ∈ A, 0 otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fU(u) = 1 2 e−|u| for u ∈ R.
6.6 Conditional density functions 95 Exercise 6.55 Let X and Y be random variables with joint density function f (x, y) =(1 4 e−1 2 (x+y) if x, y > 0, 0 otherwise.
Exercise 6.54 Let X and Y be independent random variables, each having the normal distribution with mean μ and variance σ 2. Find the joint density function of U = X − Y and V = X + Y. Are U and V independent?
Exercise 6.55 Let X and Y be random variables with joint density function f (x, y) =(1 4 e−1 2 (x+y) if x, y > 0, 0 otherwise.Show that the joint density function of U = 1 2 (X − Y ) and V = Y is fU,V (u, v) =(1 2 e−u−v if (u, v) ∈ A, 0 otherwise, where A is a region of the (u, v) plane
6.6 Conditional density functions 95
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