Let X and Y be two discrete random variables with a common mgf of et After...
9.28 Let X and Y be independent standard normal random variables. Find the mgf of X2 +Y2. What can you conclude about the distribution of X2 +Y2? (Hint: See Example 9.19.)
Let X and Y be two discrete random independent random variables. p(x) = 1/3 for x =-2,-1,0 p(y) = 1/2 for y =1,6 K = X + Y
Let X and Y be two discrete random independent random variables. p(x) = 1/3 for x =-2,-1,0 p(y) = 1/2 for y =1,6 Z = X + Y. What is the distribution of Z using the method of MGF's
2. Let X and Y be two random variables with a joint distribution (discrete or continuous). Prove that Cov(X,Y)= E(XY) - E(X)E(Y). (15 points) 3. Explain in detail how we can derive the formula Var(X) = E(X) - * from the formula in Problem 2 above. (Please do not use any other method of proof.) (10 points)
Proposition 6.10 Independent Discrete Random Variables: Bivariate Case Let X andY be two discrete random variables defined on the same sample space. Then X and Y are independent if and only if pxy(x,y) = px(x)py(y), for all x , y ER. (6.19) In words, two discrete random variables are independent if and only if their joint equals the product of their marginal PMFs. Proposition 6.11 Independence and Conditional Distributions Discrete random variables X and Y are independent if and only...
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete) and Y (continuous), for x = 1, 2, 3, 4 and 0 <y < 2. (a) Determine the marginal pmf of X. (b) Determine the marginal pdf of Y. (c) Compute P(X<2 and Y < 1). (d) Explain why X and Y are dependent without computing Cou(X,Y).
2. Let X and Y be two independent discrete random variables with the probability mass functions PX- = i) = (e-1)e-i and P(Y = j-11' for i,j = 1, 2, Let {Uni2 1} of i.i.d. uniform random variables on [0, 1]. Assume the sequence {U i independent of X and Y. Define M-max(UhUn Ud. Find the distribution
4·Let X and Y be two discrete random variables with joint density function given by Compute the probability of the following events ess than2 (b) X is even. (c) XY is even. (d) Y is odd, given that X is odd.
1. Let X and Y be two discrete random variables each with the same the possible outcomes {1,2,3} a) Construct a bivariate probability mass function Px.y : {1,2,3} x {1,2,3} + R that satisfies the following properties propeties: (i) The expectation of X is E[X] = 2.1, (ii) The conditional expectation of Y given 2 = 3 is EY 2 = 3] = 1, (iii) The correlation between X and Y is slightly positive so that 0 < corr(X,Y) <...
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].