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Problem 3. Let X and Y be two independent random variables taking nonnegative integer values (a)...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
1.9 Let Xi, -.. .Xn be nonnegative integer-valued random variables with identical pffx (-). A discrete...
1.9 Let Xi, -.. .Xn be nonnegative integer-valued random variables with identical pffx (-). A discrete mixture distribution W is created with pf fw (x)-puxi(x) +..+pfx, (x), where pi0 for i -1,... .n and X\-iPi1. Another random variable Y is defined by Y - (a) Compare the mean of W and Y. (b) If Xi,.. ,Xn are independent, compare the variance of W and Y.
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
Let X and Y be two discrete random independent random variables. p(x) = 1/3 for x...
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
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The...
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The probability distribution function is p(x) = Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ∼ N(0, 1). X and Y are independent. Meanwhile, let Z = XY . (a) What is the Expectation (mean value) of X? (b) Are Y and Z independent? (Just clarify, do not need to prove) (c) Show that Z is also a standard...
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write...
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3)...
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 .
Please select 2 & 3 2. Let X and Y be discrete random variables taking values...
Please select 2 & 3
2. Let X and Y be discrete random variables taking values 0 or 1 only, and let pr(X = i, Y = j)-pij (jz 1,0;j = 1,0). Prove that X and Y are independent if and only if cov[X,Y) 0 3. If X is a random variable with a density function symmetric about zero and having zero mean, prove that cov[X, X2] 0.
Let X and Y be independent binomial random variables B(n,p) on the same sample space. Show that X + Y is also a binomial...
Let X and Y be independent binomial random variables B(n,p) on the same sample space. Show that X + Y is also a binomial random variable B(?,?). What values should replace the questions marks?
Question 1 、 Let X, Y and Z be three random variables that take values in the alphabet {0,1, M-l...
Question 1 、 Let X, Y and Z be three random variables that take values in the alphabet {0,1, M-lj. We assume X and Z are independent and Y = X +2(mod M), The distribution of Z is given as P(Z 0)1 -p and P (Z =i)= , for i = 1, M-1. For question 1-3 we M-1 will assume that X is uniform on f0,1,..,M-1}. Find H(X) and H(Z) Find H(Y ) Find 1 (X; Y) and「X, YZ) and...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
1.9 Let Xi, -.. .Xn be nonnegative integer-valued random variables with identical pffx (-). A discrete mixture distribution W is created with pf fw (x)-puxi(x) +..+pfx, (x), where pi0 for i -1,... .n and X\-iPi1. Another random variable Y is defined by Y - (a) Compare the mean of W and Y. (b) If Xi,.. ,Xn are independent, compare the variance of W and Y.
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
Please select 2 & 3
2. Let X and Y be discrete random variables taking values 0 or 1 only, and let pr(X = i, Y = j)-pij (jz 1,0;j = 1,0). Prove that X and Y are independent if and only if cov[X,Y) 0 3. If X is a random variable with a density function symmetric about zero and having zero mean, prove that cov[X, X2] 0.
Question 1 、 Let X, Y and Z be three random variables that take values in the alphabet {0,1, M-lj. We assume X and Z are independent and Y = X +2(mod M), The distribution of Z is given as P(Z 0)1 -p and P (Z =i)= , for i = 1, M-1. For question 1-3 we M-1 will assume that X is uniform on f0,1,..,M-1}. Find H(X) and H(Z) Find H(Y ) Find 1 (X; Y) and「X, YZ) and...
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