suppose Xi, X2, . . . , X, are i.id. random variables with Xi ~ exp(A)....
Suppose Xi, X2, . . . , xn are i.id. random variables with Xi ~「α, β). Find the distribution of the sum of the X,'s and the distribution of the average of the X,'s.
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).
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. Suppose that X, , ,x, are i.id. randon variables and let X-n Σηί Xi. (a) Show that the sum of residuals always equal to zero, .e., show that (b) Show that (c) Please answer the following questions and provide a brief explanation to your answers What is the degree of freedom of (X1,.,Xn)? What is the degree of freedom of (Xi,... ,Xn, X) ? What is the degree of freedom of the residuals: (Xi -X,... ,X.-X)? What is the...
4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let X(i) < X(2) < < X(n) be their order statistics. Define Yǐ = nX(1) and Ya = (n +1 - k)(Xh) Xk-n) for 1 < k Sn. Find the joint probability density function of y, . . . , h. Are they independent? 15In
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,...
explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. (i) Write down expression for (i) E{E:-aiX.) and (ii) Var(Σ-lai%). (i) Rewrite the expression if X,'s are not independent.
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 3 1 0.5 1 E [X]-μ | 0 | and var(X)=Σー| 0 0.5 | com pute: 2 c) var(X2-X3 (d) var(X2 + X3) (e) cov(4X2 +X1,3Xi - 2X3)
Problem 1.28. Let Xi, . . . , X, be i.id. Normal(μ, σ2) random variables What is the distribution of (X+-+X,-na)/Vnơ2? How does the central limit theorem work in this case?