number2 how to solve it?
Are x1 and x2 independent
number2 how to solve it? Are x1 and x2 independent - yes, they are independent. Random variables X and Y having t...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
The joint density of random variables X1, X2 is given by fx1,x2 (x1, 2)= 6x1, for 0 < xı < 1, 0 2 <1 - r Let Y X1X2. Find the joint density of Yi and Y2 Х1, Y?
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...
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
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...
(12 points) The random variables X1, X2, and X; are jointly Gaussian with the following mean vector and covariance matrix: 54 2 07 2 5 -1 0-1 The random variable Y is formed from X1, X2, and X; as follows: Y=X1 - X2 + X3 +4. Determine P( Y> 3).
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
4 points) Let X1, X2 be independent random variables, with X1 uniform on (3,9) and X2 uniform on (3, 12). Find the joint density of Y = X/X2 and Z = Xi X2 on the support of Y, Z. f(y, z) =