Let X and Y be independent random variables
P(X=i)=P(Y=i)=1/(2^i) (i=1,2,....)
1.P(X divides Y)
=sigma( (i=1,infinite) (2^i)*((2^i)-1))
2. P((X/Y)>=1)=2/3 and for some natural number k,
P((X/Y)>=k)=2/(2^(1+k)-1)
Let X and Y be independent random variables P(X=i)=P(Y=i)=1/(2^i) (i=1,2,....) 1.P(X divides Y) =sigma( (i=1,infinite) (2^i)*((2^i)-1))...
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
8. Let X (i-1,2) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2 )2/( X2 -X1)2 < c ) =.90 b. Find P(2 X1 -3 X2< 1.5) c. Find 95th percentile of the distribution of Y-2 X1 -3 X2
Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y?
Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y?
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
Let Xi, 1-1,2, , be independent Bernoulli() random variables and let Y,-ל 1-Xi. Use the delta method to find the limiting distribution of g(%)-YAI-%), for p # 2. 1
Let X and Y be a random variables taking values 1,2, and 3 with joint I/ 01/8 나.rapis. |probabilitiespxY (1, J) given by the matrix shown: 0 1/2 0 1/8 0 1/8 4 pt. Calculate and sketch joint CDF Fxy(i,j). Find px (i) and pr(i) for ii12, 3. Compute P(X2Y). 3 pt
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
Suppose that X and Y are independent, identically distributed, geometric random variables with parameter p. Show that P(X = i|X + Y = n) = 1/(n-1) , for i = 1,2,...,n-1
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .