Question: A random experiment is to roll a dice. The corresponding random variable X is defined as follows. X = i when a number i appears.

A random experiment is to roll a dice. The corresponding random variable X is defined

as follows. X = i when a mumber i appears. i can be 1,2, 3, 4, 5,6. The dice is fair. (a) Draw probability mass function

(b) What is the marginal probability of Z? (c) Compute E(W) and E(Z) respectively.

as follows. X = i when a number i appears. i can be 1,2,3,4,5,6. The dice is fair. (a) Draw probability mass function. (b) Draw distribution function(i.e., Cumulative distribution function) (e) What is the expectation of X? 2. A random variable y follows a uniform distribution on [0, 1], It means that the random variable Y takes any value between 0 to 1. (a) What is the probability density function of Y, fy (y)? (b) What is the distribution function of T, Fy(y) = Pr[Y y]? (e) What is the expectation of Y? (d) What is the 40 percentile of Y? 3. Suppose that there are two discrete random variables, W and Z Table 1: A joint probability of W and Z Z=0 Z=1 W = 0 1/12 1/6 W = 1 2/3 1/12 (a) What is the marginal probability of W?(that is, compute Pr[W= w] for all w = 0, 1.)

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