Consider a random variable X with RX = {−1, 0, 1} and PMF P(X =
−1) = 1/4 , P(X = 0) = 1/2 , P(X = 1) = 1/4 .
a) Determine the moment-generating function (MGF) MX(t) of
X.
b) Obtain the first two derivatives of the MGF to compute E[X]
and Var(X).
Consider a random variable X with Rx = {-1,0,1} and PMF Determine the moment-generating function (MGF) Mx(t) of X b) Obtain the first two derivatives of...
Consider the following joint pmf of random variables X and Y. Y = 0 Y = 1 X = 0 1/6 1/4 X = 1 1/4 1/3 Find the marginal pmfs of X and Y. Are the random variables X and Y independent?
271 Exercise 1/1.4. Consider the joint pmf p(x, y) = cxy. 1 sxsy <3. (a) Find the normalizing constant c. (b) Are X and Y independent? Prove your claim. (c) Find the expectations of X, Y, XY.
Consider the following pmf: p(x)- .25 for x - 1, 2, 3, 4 Determine the variance of the random variable, X. C. 15/12 O infinity 8/12 O9/12
2. The joint pmf of X and Y is given below. f(x,y) 0 1 2 Y 0 1/10 3/100 1 3/10 2/10 1/10 a. Compute P(Y = 0|1 < X < 2). b. Compute E(X|Y = y) for y = 0,1. C. Evaluate Ey[E(X|Y)] using the formula Ey [E(X|Y)] = {y E(X|Y = y) f (y) and the results of part (b). d. Evaluate E(X) using the formula E(X) = Exxfx(x). Note that your answers in (c) and (d) should...
(1) Suppose the following is the joint PMF of random variables X and Y P(X x,Y y) c(3x + y), x1,2, y 1,2 where c is an unknown constant a. What is the value of c that makes this a valid joint PMF? b. Find Cov(X, Y)
(17) Suppose X has the following pmf: p(0) = 0:2; p(1) = 0:5; p(2) = 0:3. Calculate E(X), E(X2), and E(X2) - E(X)2.
1. Consider a discrete bivariate random variable (X,Y) with the joint pmf given by the table: Y X 1 2 4 1 0 0.1 0.05 2 0.2 0.05 0 4 0.1 0 0.05 8 0.3 0.15 0 Table 0.1: p(, y) a) Find marginal distributions of X and Y, p(x) and pay respectively. b) Find the covariance and the correlation between X and Y.
81. Consider the function g(x, y) given by, 1 1.52.53 11/4 0 0 0 2 0 1/8 0 0 y 3 0 1/4 0 0 4 0 0 1/4 0 5 00 0 1/8 and complete / determine the following: (a) Show that g(x, y) satisfies the properties of a joint pmf. (See Table in ?6.0.1.) (b) P(X < 2.5,Y < 3) (c) P(X < 2.5) (d) P(Y < 3) (e) P(X> 1.8, Y> 4.7) (f) E[X], EY], Var(X), Var(Y)...
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).