Let X1 , X2 , and X3 be independent and uniformly distributed
between -2 and 2.
(a) Find the CDF and PDF of Y =X1 + 2X2 .
(b) Find the CDF of Z = Y + X3 .
(c) Find the joint PDF of Y and Z . (Hint: Try the trick in Problem
2(b))
Let X1 , X2 , and X3 be independent and uniformly distributed between -2 and 2....
Let X1 , X, , and X3 be independent and uniformly distributed between-2 and 2. (a) Find the CDF and PDF ofYX, +2X2 (b) Find the CDF of Z-), + X, . (c) Find the joint PDF of Y and Z.(: Try the trick in Problem 2(b) Let X1 , X, , and X3 be independent and uniformly distributed between-2 and 2. (a) Find the CDF and PDF ofYX, +2X2 (b) Find the CDF of Z-), + X, . (c)...
2. Let Xi exp(1) and X2 ~ variables with rate 1. Let: erp(1) be independent and identically-distributed exponential random (a) What is the cdf of X1? b) What is the joint pdf of (Xi, X2)? (c) What is the joint pdf of (Y, Z)? d) What is the marginal pdf of z?
Suppose X1, X2, Xz~exp(1) and they are independent. (a) Compute the cdf of X1 (b) Let Y- max(Xi, X2, X3). Find the cdf of Y (c) Derive the pdf of Y
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Yı, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fxı,Y,(y1, y2). iii) Suppose Y3 = X1 + X2 + X3. Compute the covariance...
Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn) uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (X1, X2,..., Xn) and N -min (X1, X2,..., Xn) 1. Find the joint distribution of a pair (M,N) 2. Derive the CDF and density for M 3. Derive the CDF and density for N.
Let X1,X2 be two independent exponential random variables with λ=1, compute the P(X1+X2<t) using the joint density function. And let Z be gamma random variable with parameters (2,1). Compute the probability that P(Z < t). And what you can find by comparing P(X1+X2<t) and P(Z < t)? And compare P(X1+X2+X3<t) Xi iid (independent and identically distributed) ~Exp(1) and P(Z < t) Z~Gamma(3,1) (You don’t have to compute) (Hint: You can use the fact that Γ(2)=1, Γ(3)=2) Problem 2[10 points] Let...
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y? (b) Let X1, X2, X3 be independent random variables, uniformly distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is the middle of the three values). Find the conditional CDF of X1, given that Y = 0.5. Under this conditional distribution, is...
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.