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HW 5 problems...

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ENEE324 (Fall 2020): Professor Shayman Assignment 5 1. Let X be a discrete random variable that takes values −1, 0, 1, 2 with probabilities 0.2, 0.1, 0.4, 0.3, respectively. Let Y = X 2 + 4. (a) Determine the pmf of Y . (b) Determine the expected value of Y . (c) Determine the variance of Y . 2. Let X and Y be random variables (on the same sample space). The joint pmf is given by the following table:

Y=0 Y=1

X=0 0.2 0.1

X=1 0.3 0

X=2 0.1 0.3

(a) Determine the pmf of X . (b) Determine the conditional pmf of X given Y = 0. (c) Determine the conditional mean (expected value) of X given Y = 0. (d) Determine the probability that X is greater than Y . (e) Determine the probability that X is greater than Y given that X is even. 3. Let X and Y be random variables (on the same sample space). The joint pmf is given by the table below. Let Z = X 2 − 2Y .

Y=0 Y=1

X=0 0.2 0.2

X=1 0.3 0.1

X=2 0.2 0

(a) Determine the pmf of X . (b) Determine the conditional pmf of X given Y = 0. (c) Determine the conditional mean (conditional expectation) of X given Y = 0. (d) Determine P (2X − Y > 1) (e) Determine the expected value of Z . (f) Determine the variance of Z . 4. A fair coin is tossed three times. Let X = 0 if the first toss results in tails, and X = 1 if the first toss results in heads. Let Y denote the total number of heads obtained in the three tosses.

(a) Determine the joint pmf of X and Y . (b) Determine the conditional pmf of X given Y = 1. (c) Determine the conditional pmf of Y given X = 1. (d) Determine P (X < Y ). 5. The random variable X is uniformly distributed in the interval [−1, 2]. (a) Find and plot the cdf of X . (b) Use the cdf to find the probabilities of the following events: A = {X ≤ 0}, B = {|X − 0.5| < 1}, and C = {X > −0.5}. 6. The cdf of the random variable X is given by  0 x < −1     0.5 −1 ≤ x < 0 FX (x) = (1 + x)/2 0 ≤ x < 1    1 x≥1

(a) Plot the cdf and identify the type of random variable. (b) Find P [X ≤ −1], P [X = −1], P [X < 0.5], P [−0.5 < X < 0.5], P [X > −1], P [X ≤ 2], P [X > 3]. 7. A random variable X has cdf: ( 0 FX (x) = 1 − 41 e−2x

x 10]. 8. A random variable X has pdf: ( cx(1 − x2 ) 0 ≤ x ≤ 1 fX (x) = 0 otherwise (a) Find c and plot the pdf. (b) Plot the cdf of X . (c) Find P [0 < X < 0.5], P [X = 1], P [0.25 < X < 0.5].

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