1 Exercises for lecture 1

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1 1 Exercises for lecture 1 Exercise 1 a) Show that if F is symmetric with respect to µ, and E( X ) <, then E(X) = µ. Moreover, if F admits an unimodal density, then the mean = mediane = mode. b) If F is symmetric and all absolute moments µ k exist, then the moments µ k = 0 for all odd k. If F is symmetric with respect to µ and all the moments µ k exist, then µ k = 0 for all odd k (e.g., µ 3 = 0). Exercise 2 Provide an example of asymmetric density with α = 0. Exercise 3 Show that µ 4 /σ4 = 3 and β = 0 for normal distribution N(µ, σ 2 ). Exercise 4 Let (ξ n ) and (η n ) be two sequences of r.v.. Prove the following statements: 1 o. If a R is a constant then ξ n D a ξn P a, when n. 2 o. (Slutsky s theorem) If ξ n D a and ηn D η when n and a R is a constant then ξ n + η n D a + η, as n. Show that if a is a general r.v., these two relations do not hold (construct a counterexample). 3 o. Let ξ n P a, and let ηn D η when n, where a R is a constant and η is a random variable, Then ξ n η n D aη, as n. Would this result continue to hold if we suppose that a is a general random variable? Exercise 5 A r.v. Y takes values 1, 3 and 4 with the probabilities P (Y = 1) = 3/5, P (Y = 3) = 1/5 et P (Y = 4) = 1/5. How would you generate Y given a r.v. U U(0, 1). Exercise 6 Let U U(0, 1). 1. Explain how to simulate a dice with 6 faces given U. 2. Let Y = [6U + 1], where [a] is the integer part of a. What are possible values of Y and the corresponding probabilities? 1

2 Exercise 7 Suppose 2 balanced dices are drawn. Find joint probability distribution of X and Y if: 1. X is the maximum of the obtained values and Y is a sum; 2. X is the value of the first dice and Y is the maximum of the two; 3. X and Y are, respectively, the smallest and the largest value. Exercise 8 Suppose that X and Y are 2 independent Bernoulli B( 1 2 ) random variables. Let U = X + Y and V = X Y. 1. What is the joint probability distribution and marginal probability distributions of U and V, conditional distribution of U given V = 0 et V = are r.v. U and V independent? Exercise 9 Let ξ 1,..., ξ n be independent r.v., and let 1) Show that ξ min = min(ξ 1,..., ξ n ), ξ max = max(ξ 1,..., ξ n ). n P (ξ min x) = P (ξ i x), n P (ξ max < x) = P (ξ i < x). 2) Suppose, furthermore, that ξ 1,..., ξ n are identically distributed with uniform distribution U[0, a]. Compute E(ξ min ), E(ξ max ), Var(ξ min ) et Var(ξ max ) Exercise 10 Let ξ 1,..., ξ n be i.i.d. Bernoulli r.v. with P (ξ 1 = 0) = 1 λ i, where λ i > 0 and > 0 is small. Show that ( n ) ( n ) P ξ i = 1 = λ i + O( 2 ), P Exercise 11 P (ξ 1 = 1) = λ i ( n ξ i > 1 1) Prove that inf <a< E((ξ a) 2 ) is attained for a = E(ξ) and so inf E((ξ <a< a)2 ) = Var(ξ). ) = O( 2 ). 2) Let ξ be a nonnegative r.v. with c.d.f. F and finite expectation. Prove that E(ξ) = 0 (1 F (x))dx. 3) Show, using the result of 2), that if M is the median of the c.d.f. F of ξ, inf E( ξ a ) = E( ξ M ). <a< 2

3 Exercise 12 Let X 1 and X 2 be two independent r.v. with the exponential distribution E(λ). Show that min(x 1, X 2 ) and X 1 X 2 are r.v. with distributions, respectively, E(2λ) and E(λ). Exercise 13 Let X be the number of 6 in independent draws of a dice. Using the Central Limit Theorem estimate the probability that 1800 < X 2100 (Φ( 6) , Φ(2 6) ). Compare this approximation to that obtained using the Chebyshev inequality. Exercise 14 Suppose that r.v. ξ 1,..., ξ n are mutually independent and identically distributed with the c.d.f. F. For x R, let us define the random variable F n (x) = 1 n µ n, where µ n is the number of ξ 1,..., ξ n which satisfy ξ k < x. Show that for any x F n (x) P F (x) (the function F n (x) is called the empirical distribution function). Exercise 15 [Monté-Carlo method] We want to compute the integral I = 1 0 f(x)dx. Let X be a U[0, 1] random variable, then E(f(X)) = 1 0 f(x)dx = I. Let X 1,..., X n be uniformly distributed on [0, 1] i.i.d. r.v.. Let us consider the quantity f n = 1 n f(x i ) n and let us suppose that σ 2 = Var(f(X)) <. Prove that E( f n ) I et f n P I as n. Estimate P ( f n I < ɛ) using the CLT. Exercise 16 Weibull distributions is often used in the survival and reliability analysis. distribution from this family is given by c.d.f. { 0, x < 0 F (x) = 1 e 5x2, x 0. An example of a Explain how to generate a r.v. Z F given a uniform r.v. U. Exercise 17 Write down the algorithm of simulating a Poisson r.v. by inversion. Hint: there is no simple expression for the Poisson c.d.f. and the set of values is infinite. However, the Poisson c.d.f. can be easily computed recursively. Observe that if X is the poisson r.v., λ λk P (X = k) = e k! = λ P (X = k 1). k 3

4 2 Exercises for lecture 2 Exercise 18 We have X and Z, 2 r.v., independent with exponential distribution, X E(λ), Z E(1). Let Y = X + Z. Compute the regression function g(y) = E(X Y = y). Exercise 19 Suppose that the joint distribution of X and Y is given by { 1 e F (x, y) = 2x e y + e (2x+y) si x > 0, y > 0, 0 sinon. 1. Find the marginal distribution of X and Y. 2. Find the joint density of X and Y. 3. Compute the marginal densities of X and Y, conditional density of X given Y = y. 4. Are X and Y independent? Exercise 20 Consider the joint density function of X and Y given by: 1. Verify that f is a joint density. f(x, y) = 6 7 (x2 + xy ), 0 x 1, 0 y Find the density of X, the conditional density f Y X (y x). ( ) 3. Compute P Y > 1 2 X < 1 2. Exercise 21 The joint density X and Y is given by: f(x, y) = e (x+y), 0 x <, 0 y < Compute 1. P (X < Y ); 2. P (X < a). Exercise 22 Two point are chosen at random on the opposite sides of the middle point the interval of length L. In other words, the two points X and Y are independent random variables such that X is uniformly distributed over [0, L/2[, and Y is uniformly distributed over [L/2, L]. Find the probability that the distance X Y is larger than L/3. 4

5 Exercise 23 Let U 1 and U 2 be 2 independent r.v., both being uniformly distributed on [0, a]. Let V = min{u 1, U 2 } and Z = max{u 1, U 2 }. Show that the joint c.d.f. F of V and Z is given by F (s, t) = P (V s, Z t) = t2 (t s) 2 a 2 for 0 s t a. Hint: note that V s and Z t iff both U 1 t and U 2 t, but not both s < U 1 t and s < U 2 t. Exercise 24 Given 2 independent r.v. X 1 and X 2 with exponential distribution with parameters λ 1 and λ 2, find the distribution of Z = X 1 /X 2. Compute P (X 1 < X 2 ). Exercise 25 Let X and Y be i.i.d. r.v.. Use the definition of the conditional expectation to show that E(X X + Y ) = E(Y X + Y ) (p.s.), and thus E(X X + Y ) = E(Y X + Y ) = X+Y 2 (a.s.). Exercise 26 Let X, Y 1 and Y 2 be independent r.v., let Y 1 and Y 2 be normal N(0, 1), et Z = Y 1 + XY X 2 Using the conditional distribution P (Z < u X = x) show that Z N(0, 1). Exercise 27 Let X and Y be 2 square integrable r.v. on (Ω, F, P ). Prove that Exercise 28 Var(Y ) = E(Var(Y X)) + Var(E(Y X)). Let X 1,..., X n be independent r.v. such that X i P(λ i ) (Poisson distribution with parameter λ i, i.e. P (X i = k) = e λ i λk i k! ). 1 o. Find the distribution of X = n X i. 2 o. Show that the conditional distribution of (X 1,..., X n ) given X = r is multinomial M(r, p 1,..., p n ) (you will compute the corresponding parameters). Recall that r.v. (X 1,..., X k ) integer valued in {0,..., r} have multinomial distribution M(r, p 1,..., p k ) if r! P (X 1 = n 1,..., X k = n k ) = n 1!...n k! pn pn k k, with k n i = r. This is the distribution of (X 1,..., X k ) where X i = number of Y s which are equal to i in r independent trials Y 1,..., Y r with probabilities P (Y 1 = i) = p i, i = 1,..., k. Note that if k = 2, P (X 1 = n 1, X 2 = r n 1 ) = P (X 1 = n 1 ), and the distribution is denoted M(r, p) = B(r, p). 3 o. Compute E(X 1 X 1 + X 2 ). 4 o. Show that if X n is binomially distributed, X n B(n, λ/n), then for all k, P (X n = k) tends to e λ λk k! when n. 5

6 Recall trials, that binomial distribution describes the number X of wins in n independent Bernoulli P (X = k) = C k np k (1 p) n k. Exercise 29 Show that 1. Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z), ( n 2. Cov X i, ) n j=1 Y j = n nj=1 Cov(X i, Y j ). 3. Prove that if Var(X i ) = σ 2 and Cov(X i, X j ) = γ for all 1 i, j n then Var(X X n ) = nσ 2 + n(n 1)γ. 4. Let ξ 1 and ξ 2 be i.i.d. random variables such that 0 < Var(ξ 1 ) <. Show that r.v. η 1 = ξ 1 ξ 2 and η 2 = ξ 1 + ξ 2 are uncorrelated. Exercise 30 Let X be the number of 1 and Y the number of 2 in n draws of a honest (balanced) dice. Compute Cov(X, Y ). Before computing the quantity, would you be able to predict if Cov(X, Y ) 0 or Cov(X, Y ) < 0. Hint: use the relationship 2) of Exercise 29. Exercise 31 1 o. Let ξ and η be r.v. with E(ξ) = E(η) = 0, Var(ξ) = Var(η) = 1 and the correlation coefficient ρ. Show that E(max(ξ 2, η 2 )) ρ 2. Hint: observe that max(ξ 2, η 2 ) = ξ2 + η 2 + ξ 2 η o. Let ρ be the correlation of η and ξ. Show the inequality Exercise 32 P ( ) ξ E(ξ) ɛ Var(ξ) or η E(η) ɛ Var(η) ρ 2 ɛ 2. Let (X, Y ) be a random vector of dimension 2. distribution of X given Y = y is N(y, σ 2 ). 1 o. What is the distribution of Y given X = x? 2 o. What is the distribution of X? 3 o. What is the distribution f E(Y X)? Suppose that Y N(m, τ 2 ) and that the Exercise 33 6

7 Let X and N be r.v. such that N is valued in {1, 2,...} and E( X ) <, E(N) <. Consider the sequence X 1, X 2,... of independent r.v. with the same distribution as X. Show the Wald identity: if N is independent of X i then Exercise 34 N E( X i ) = E(N)E(X). Suppose that the salary of an individual satisfies Y = Xb + σε, where σ > 0, b R, X is a r.v. with bounded second order moments corresponding the capacities of the individual, and ε is independent of X standard normal variable, ε N(0, 1). If Y is larger than the SMIC value S, the received salary Y is Y, otherwise it is equal to S. Compute E(Y X). Is this expectation linear? Exercise 35 Show that if φ is a characteristic function of some r.v. then φ, φ 2 and Re(φ), are also characteristic functions (of certain r.v.). Hint: for Re(φ) consider 2 independent random variables X and Y, where Y takes values 1 and 1 with probabilities 1/2, X has characteristic function φ, then compute the characteristic function of XY. 7

8 3 Exercises for lecture 3 Exercise 36 Two random vectors ( X ) R p and Y R q are independent( iff the ) characteristic function φ Z (u) X a of the vector Z = can be represented, for any u =, a R Y b p and b R q, as φ Z (u) = φ X (a)φ Y (b). Verify this characterization in the continuous case. Exercise 37 Let the joint density of r.v. s X and Y satisfy f(x, y) = 1 What is the distribution of X, of Y? Exercise 38 2π e x 2 2 e y2 2 [1 + xyi{ 1 x, y 1}], Prove that any linear transformation of a normal vector is again a normal vector: if Y = AX +c where A R q p and c R q are some fixed matrix and vector (non-random), Exercise 39 Compute the density of the χ 2 p distribution. Exercise 40 Y N q (Aµ + c, AΣA T ). Compute the density of the Fisher F p,q distribution. Exercise 41 Compute the density of the Student t q distribution. Exercise 42 Let Q be a q p matrix with q > p of rank p. 1 o. Show that P = Q(Q T Q) 1 Q T is a projector. 2 o. On what subspace L does P project? Exercise 43 Let (X, Y ) be a random vector with density f(x, y) = C exp( x 2 + xy y 2 /2). 1 o. Show that (X, Y ) is a normal vector. Compute the expectation, the covariance matrix and the characteristic function of (X, Y ). Compute the correlation coefficient ρ XY of X and Y. 2 o. What is the distribution of X? of Y? of 2X Y? 3 o. Show that X and Y X are independent random variables with the same distribution. 8

9 Exercise 44 Let X N(0, 1) and let Z be a r.v. taking values 1 and 1 with probability 1 2. We suppose that X and Z are independent, we set Y = ZX. 1 o. Show that the distribution of Y is N(0, 1). 2 o. Compute the covariance and the correlation of X and Y. 3 o. Compute P (X + Y = 0). 4 o. Is (X, Y ) a normal vector? Exercise 45 Let ξ and η be independent r.v. with uniform distribution U[0, 1]. Prove that X = 2 ln ξ cos(2πη), Y = 2 ln ξ sin(2πη) satisfy Z = (X, Y ) T N 2 (0, I). Hint: let (X, Y ) N 2 (0, I). Change to the polar coordinates. Exercise 46 Let Z = (Z 1, Z 2, Z 3 ) T be a normal vector, with density f, f(z 1, z 2, z 3 ) = ( 1 exp 6z z z z ) 1z 2. 4(2π) 3/ o. What is the distribution of (Z 2, Z 3 ) given Z 1 = z 1? Let X and Y be random vectors defined with X = Z et Y = ( o. Vector (X, Y ) of dimension 6, is it Gaussian? Does vector X have a density? Does vector Y have a density? 3 o. Are vectors X and Y independent? 4 o. What are the distributions of the components of Z. Exercise 47 Let (X, Y, Z) T be a normal vector with zero mean and covariance matrix Σ = o. We set U = X + Y + Z, V = X Y + Z, W = X + Y Z. What is the distribution of the vector (U, V, W ) T? 2 o. What is the density of the random variable T = U 2 + V 2 + W 2? Exercise 48. ) Z. 9

10 Let a vector (X, Y ) be normal N 2 (µ, Σ) with mean and covariance matrix: ( ) ( ) µ =, Σ = o. What is the distribution of X + 4Y? 2 o. What is the joint distribution of Y 2X and X + 4Y. Exercise 49 Let X be a zero mean normal vector of dimension n with covariance matrix Σ > 0. What is the distribution of the r.v. X T Σ 1 X? Exercise 50 We model the height H of a male person in population P by the Gaussian distribution N(172, 49) (units: cm). In this model: 1 o. What is the probability for a man to be of height 160cm? 2 o. We assume that there are approximately 15 millions of adult men in P; provide an estimation of a number of men of height 200cm. 3 o. What is the probability for 10 men randomly drawn from P to be all in the interval [168,188]cm? The height H of females of P is modeled by the Gaussian distribution N(162, 49) (units: cm). 4 o. What is the probability for a male chosen at random to be higher than a randomly chosen female? We model the heights (H, H ) of a man and a woman in a couple by a normal vector, the correlation coefficient ρ between the height of a woman and a man being 0.4 (respectively, 0.4). 5 o. Compute the probability p (respectively, p ) that the height of a man in a couple is larger than that of a woman (before making the computation, what would be your guess, in which order should one range p and p?). Exercise 51 Let Y = (η 1,..., η n ) T be a normal vector, Y N n (µ, σ 2 I), H n J be a subspace of dimension n J J > 0 of R n, and let H n J M be a subspace of dimension n J M, M > 0 of H n J. We set d J = min Y y et d J+M = min Y y. y H n J y H n J M Verify that distribution (with J degrees of free- 1. if µ H n J then random variable d 2 J /σ2 follows χ 2 J dom); 2. if, in addition, µ H n J M, then J d 2 J+M d2 J M d 2 J F M,J (Fisher distribution with (M, J) degrees of freedom). 10