TOPIC:Properties of Expectation.
Let X1 and X2 be two discrete random variables with joint p.m.f. P(X1k1,X2 - k2). Prove...
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...
Let X1 and X2 be two discrete random variables, where X1 can attain values 1, 2, and 3, and X2 can attain values 2, 3 and 4. The joint probability mass function of these two random variables are given in the table below: X2 X1 2 3 4 1 0.05 0.04 0.06 2 0.1 0.15 0.2 3 0.2 0.1 0.1 a. Find the marginal probability mass functions fX1 (s) and fX2 (t). b. What is the expected values of X1...
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 and X2 be random variables, not necessarily independent. Show that E [X1 + X2] = E [X1] + E [X2]. You may assume that X1 and X2 are discrete with a joint probability mass function for this problem, while the above inequality is true also for continuous random variables.
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?
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) =
Exercise 11. Let Xi,Y be random variables with joint PDF fxi.Y. Let X2,Y be random variables with joint PDF fXyXy Let T: R2 → R2 and let S: R2 → R2 so that ST(x,y) = (z, y) and TS(z, y)-(x,y) for every (x,y) є R2. Let J(z, y) denote the determinant of the Jacobian of S at (x,y). Assume that (X2,Y) = T(X1Ύǐ). Using the change of variables formula from multivariable calculus, show that fx2 x2 (x, y)-fx .yi (S(x,...
Consider two discrete random variables X and Y with joint p.m.f given below . 0.200.15 -1 0 0.05 0.15 0.10 1 0.200.15 Find the joint p.m.f. of U-X Y and V X Yand enter it below Pulu,y) -1 0 -1 0 Chec
Exercise 11. Let Xi,y, be random variables with joint PDF fXiXi. Let X2,Y2 be random variables with joint PDF fx2,Y2. Let T: R2R2 and let S: R2 -R2 so that ST(x, y) (z, y) and TS(a, y) (x, y) for every (x, y) E R2. Let J(x, y) denote the determinant of the Jacobian of S at (x,y. Using the change of variables formula from multivariable calculus, show that
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)