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Question Description

I need help with 2 statistics questions

Problem 4. Consider two independent discrete random variables:

Define the random variable X as the number rolled on a fair, six-sided die. So, the sample space for X

is {1, 2, 3, 4, 5, 6} and all of the six outcomes are equally likely.

Define the random variable Y as a binomial random variable with n = 1 trial and success probability

p = 0.75. That is Y ∼ Binom(1, 0.75).

Define a new random variable W = X − Y

(a) Find the sample space and probability mass function for W . Hint: keep in mind there may be more than one path to a single sample space value. For example, W=1 can be achieved through 2 disjoint possibilities for (X, Y ): (1, 0) or (2, 1). (10pts)

(b) Find the expected value of W. (5pts)

Problem 3. Consider the following function: x

4 0 ≤ x ≤ 2

f(x)= C 2<x≤3 0 otherwise

(a) Find the value C such that the above function is a valid probability density function. (6pts)

(b) Let X be a continuous random variable characterised by the above probability density function. Find Pr(−1 ≤ X ≤ 1). (4pts)

(c) Suppose we sample 5 random values for X; that is, 5 values for X are drawn from the same probability distribution defined by the PDF above, all independent of each other. What is the probability a majority of these observations are less than 2? (6pts)