Question 1

Let 𝑋𝑋 and 𝑌𝑌 be discrete random variables with joint frequencies shown in Table 1 below

(a) With clear workings, evaluate (i.e. compute) the following probabilities: i) 𝑃𝑃(𝑋𝑋 >1, 𝑌𝑌 = 3 ), ii) 𝑃𝑃(𝑋𝑋 ≥ 2, 𝑌𝑌2 < 4 ).

(b) Compute

𝑃𝑃(𝑋𝑋 = 𝑗𝑗)𝑃𝑃(𝑌𝑌 = 1|𝑋𝑋 = 𝑗𝑗) 𝑗𝑗=0 and
𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 0) + 𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 1) + 𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 2) + 𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 3)

What do you observe about the two terms? Explain the rationale behind your
observation(s).

(c) Discuss the steps you would implement to calculate 𝐸𝐸[𝑌𝑌|𝑋𝑋 ≤ 1]. What are some practical applications of conditional expectations?

Question 2

Let 𝜙𝜙 and Φ be the pdf and cdf, respectively, of the standard normal variable 𝑍𝑍. Let 𝑋𝑋 = 2𝑍𝑍 − 3. Let 𝑌𝑌 be a binomial random variable, independent of 𝑋𝑋, with parameters 𝑛𝑛 = 10 and 𝑝𝑝 = 0.2.

(a) Can you determine 𝑃𝑃(𝑋𝑋 ≤ 1|𝑌𝑌 = 2)? If so, calculate this probability.

(b) Determine the probability for which 𝑋𝑋 is within one standard deviation of its mean.

(c) Appraise if we may determine the mean, median and variance of 𝑅𝑅 = 𝑋𝑋 + 𝑌𝑌. If so, compute them.

(d) Let 𝑍𝑍1, 𝑍𝑍2, … , 𝑍𝑍5 be 5 independent standard normal random variables. For each 𝑗𝑗 = 1, … ,5, define 𝑋𝑋𝑗𝑗 = 2𝑍𝑍𝑗𝑗 − 3

Let 𝑆𝑆 = 1 4��𝑋𝑋𝑗𝑗 + 3� 25 𝑗𝑗=1

Using what you understand about normal random variables and the random variables associated with it, describe how you would compute 𝑃𝑃(𝑆𝑆 > 2). Therefore, compute 𝑃𝑃(𝑌𝑌 > 2|𝑆𝑆 > 2).