2. Suppose that Y is a standard normal random variable (a) Calculate E Y3] and E...
X Y Z iid
Suppose for random variable X, P(X > a) - exp( random variable Y, P(Y > y) exp(-0y) for y > 0, and for random variable , P(Z > z)--exp(-фа) for z > 0. (a) Obtain the moment generating functions of X, Y and Z. (b) Evaluate E(X2IX > 1) and show it is equal to a quadratic function of λ. (c) Calculate P(X > Y Z) if λ-1, θ--2 and φ--3. -λα) for x > 0,...
Let X variable Y by be a normal random variable with mean 0 and variance 1. We define the random y2 if x 20, Y= (a For t E R, compute Mr()-Elen'], the moment generating function of Y. Compute EY
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint: Determine first the probability distribution of T -X +Y + Z using the moment generating function method. Moment generating function for Poisson random variable is given in earlier lecture notes)
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint:...
Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The...
a) Prove that if Y is a random variable with all of its cumulants of order greater than two equal to zero, i.e., 0 = K_3 = K_4 = ......... then Y has a normal distribution. b) Suppose that Z has cumulant generating function k_Z(t) with E(Z) = 0 and Var(Z) = 1. Let Y = σZ + μ. Find the cumulant generating function of Y , k_Y (t) in terms of k_Z(.). Use this to prove that all cumulants...
(10 points) 4. The moment generating function of a random variable Y is , for t e R, where k is a constant. (a) Find the mean of Y. (b) Determine Pr(Y <1Y <2) (c) Find th e cumulative distribution function of Y, with domain R.
(10 points) 4. The moment generating function of a random variable Y is , for t e R, where k is a constant. (a) Find the mean of Y. (b) Determine Pr(Y
(6) Suppose that X is an absolutely continuous random variable with density 1<I<2 f(3) = lo, otherwise. Find (a) the moment generating function MX(t). (b) the skewness of X (c) the kurtosis of X (7) Suppose that X, Y and Z are random variables such that p(X,Y) = 1 and p(Y,Z) = -1. What is p(X, Z)? Explain your answer. (8) Suppose that X, Y and Z are random variables such that p(X,Y) = -1 and p(Y,Z) = 0. What...
Let the random variable Y have the following probability distribution y 2 4 6 P(Y=y) 4/k 1/k 5/k find the value of k. find the moment-generating function of Y find Var(Y) using the moment generating function let W= 2Y-Y^2 +e^2*Y+7. find E(W)
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B)
(a) If var[X o2 for each Xi (i...
problems binomial random, veriable has the moment generating function, y(t)=E eux 1. A nd+ 1-p)n. Show that EIX|-np and Var(X) np(1-p) using that EIX)-v(0) nd E.X2 =ψ (0). 2. Lex X be uniformly distributed over (a b). Show that ElXI 쌓 and Var(X) = (b and second moments of this random variable where the pdf of X is (x)N of a continuous randonn variable is defined as E[X"-广.nf(z)dz. )a using the first Note that the nth moment 3. Show that...