Let X be a discrete random variable with PMF
(a) Find P(X ≤ 9).
(b) Find E[X] and Var(X).
(c) Find MX(t), where t < ln 3.

3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2 for 0.2 for x=3 Pg(x)=〈 0.1 for x=4 0.25 for x=5 0.15 for x=6 otherwise a) (10 points) Find E[X] b) (10 points) Find Var(X) c) Let Y-* I. (15 points) Find E[Y] II. (15 points) Find Var(Y) X-HX 4. Consider a discrete random variable X with E [X]-4x and Var(X) = σ. Let Y a. (10 points) Find E[Y] b. (20 points) Find...
Let X be a discrete random variable with PMF: a. Find the value of the constant K b. Find P(1 < X ≤ 3)
Let X be a discrete random variable with the following PMF. Px(k) = 1/4 for k = -2 1/8 for k = -1 1/8 for k = 0 1/4 for k = 1 1/4 for k = 2 0 otherwise Define a new random variable Y = (X + 1)2 a) Find E[X] and Var[X] b) Find the range of Y and write its PMF. c) Show that the PMF of Y is a valid PMF. d) Find P(Y ≤...
5. Let X be a discrete random variable with the following PMF: for x = 0 Px(x)- for 1 for x = 2 0 otherwise a) Find Rx, the range of the random variable X. b) Find P(X21.5). c) Find P(0<X<2). d) Find P(X-0IX<2)
Let X be a discrete random variable with the following PMF 6 for k € {-10,-9, -, -1,0, 1, ... , 9, 10} Px(k) = otherwise The random variable Y = g(X) is defined as Y = g(x) = {x if X < 0 if 0 < X <5 otherwise Calculate E[X], E[Y], var(X), and var(Y) for the two variables X and Y
Let X be a discrete random variable, and let Y X (a) Assume that the PMF of X is Ka2 0 if x- -3, -2,-1,0,1,2,3 otherwise, where K is a suitable constant. Determine the value of K. (b) For the PMF of X given in part (a) calculate the PMF of Y (c) Give a general formula for the PMF of Y in terms of the PMF of X
0.25 x-1 0.15 x2 6. Let X be a discrete random variable with PMF: Px(x) 0.2 x-3 0.1 x 4 0.3 x-5 0 otherwise a. (10 points) Find E[X] b. (5 points) Find Var(X)
Let X be a discrete random variable with a the PMF such that P(X =x)= 2/3^x for x =1, 2,3,....... Find the expected value Of X.
Let X be a discrete random variable with the following pmf 0.1 for I = 0.2 0.2 for x = 0.4 0.2 for x = 0.5 P(X = x) = 0.3 for x = 0.8 0.2 for x = 1 0 otherwise Note: Write your final answers as decimals Find the following a) P(0.25 < X < 0.75) = b) P(X = 0.2|X<0.6) c) E(2X+1) =
2. For a discrete random variable X, with CDF F(X), it is possible to show that P(a < X S b)-F(b) - F(a), for a 3 b. This is a useful fact for finding the probabil- ity that a random variable falls within a certain range. In particular, let X be a random variable with pmf p( 2 tor c-1,2 a. Find the CDF of X b. Find P(X X 5). c. Find P(X> 4). 3. Let X be a...