4. Suppose Xi, X2, X3 ~exp(1) and they are independent (a) Compute the edf of X...
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
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?
Let Xi, X2, X3 be i.id. N(0.1) Suppose Yı = Xi + X2 + X3,Ý, = Xi-X2, у,-X,-X3. Find the joint pdf of Y-(y, Ya, y), using: andom variables. a. The method of variable transformations (Jacobian), b. Multivariate normal distribution properties.
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y. (c) Let X1, X2,...,be i.i.d. Exp(1), and let Mk = max(X1,..., Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk (Hint: P(min(X1, X2, X3) > k) = P(X1 > k, X2 > k, X3 > k), how about max ?) (d) Show that as k- , the CDF of...
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk =
max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability
density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1
>= k, X2 >= k, X3 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
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))
This is a probability question. Please be thorough and
detailed.
3. (8 pts.) Suppose that Xi ~ Exp(A) and X2 ~ Exp(A2) where λ1 and λ2 are positive con- λ2, but do assume that Xi and X2 are independent. Compute stants. Do not assume λι P(X1 < X2). Now note that the probability you just computed is in fact P(Xmin(XI, X2)). This suggests the following generalization. Suppose we have a collection of N independent ex- ponential random variables, X1, X2,...
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y = max(X,Xy) and Z-X1 + X2. (1) Calculate the joint PDF of Y and Z. (2) Derive the marginal PDF of both Y and Z. Are Y and Z independent?
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y...
5. [22] If Xi, X2, and X3 are independent random variables with E(XI) 4, E(X2)-3, E(X3)2, V(X) = 1, V(X:) = 5, V(Xs) = 2, and Y = 2X1 + X2-3X1 (a) Determine E(Y) and V(Y). P(Y > 2.0) and P( 1.3 Y 8.3).
7. Suppose that Xi,..., Xk are independent random variables, and X, ~ Exp(B) for i = 1, . . . , k. Let Y = min(X1 , . . . , Xk). Show that Y ~ Exp(Σ-1 β).