University |
Singapore University of Social Science (SUSS) |

Subject |
ANL321: Statistical Methods |

^{th}Jun 2023

# ANL321: Let ππ and ππ be discrete random variables with joint frequencies shown in Table 1 below: SUSS, Assignment

**Question 1**

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

(a) With clear workings, evaluate (i.e. compute) the following probabilities: i) ππ(ππ >1, ππ = 3 ), ii) ππ(ππ β₯ 2, ππ2 < 4 ).

(b) Compute

ππ(ππ = ππ)ππ(ππ = 1|ππ = ππ) ππ=0 and

ππ(ππ = 1, ππ = 0) + ππ(ππ = 1, ππ = 1) + ππ(ππ = 1, ππ = 2) + ππ(ππ = 1, ππ = 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 πΈπΈ[ππ|ππ β€ 1]. What are some practical applications of conditional expectations?

**Question 2**

Let ππ and Ξ¦ be the pdf and cdf, respectively, of the standard normal variable ππ. Let ππ = 2ππ β 3. Let ππ be a binomial random variable, independent of ππ, with parameters ππ = 10 and ππ = 0.2.

(a) Can you determine ππ(ππ β€ 1|ππ = 2)? If so, calculate this probability.

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

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

(d) Let ππ1, ππ2, β¦ , ππ5 be 5 independent standard normal random variables. For each ππ = 1, β¦ ,5, define ππππ = 2ππππ β 3

Let ππ = 1 4οΏ½οΏ½ππππ + 3οΏ½ 25 ππ=1

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

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