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Past Exam 2

MTH2222

Throughout this paper, CDF, PMF, PDF and MGF stand for cumulative distribution function, probability mass function, probability density function and moment generating function, respectively. All answers must be justified.

1 A car insurance company classifies its policy holders as good, bad or average risks: 30% are deemed good risks, 20% are deemed bad risks, and 50% are deemed average risks. Historical data suggest that 10% of the good risks, 40% of the bad risks, and 20% of the average risks will be involved in an accident in the coming year.

(a) What is the probability that a randomly chosen customer will be involved in an accident in the coming year?

(b) An accident claim has just been filed with the company. What is the probability that this customer was classifified as good risk?            [5 marks]

2 Let X be a discrete random variable with PMF

Obtain c, E[X] and var(X).

Hint: You may use the fact that, for 0 < z < 1,

                              [8 marks]

3 Let X be an exponential random variable with parameter λ > 2.

(a) Find the PDF of Y = e X.

(b) Obtain the mean and variance of Y .                     [7 marks]

4 Suppose the PDF of X is given by

Find the MGF of X.                         [4 marks]

5 Let X be a random variable with MGF given by

Find the mean and variance of X. Hint: You may use the fact that M′′X(t) =                   [5 marks]

6 The joint PMF of X and Y is given by

(a) Describe the possible values of p and q.

(b) Find the marginal PMFs of X and Y .

(c) Find the conditional PMF of Y given X = x for x = −1, 0, 1.

(d) Find the conditional expectation of Y given X = x for x = −1, 0, 1, and deduce cov(X, Y ).

(e) Are X and Y independent? Explain.                                 [13 marks]

7 The joint PDF of X and Y is given by

where n is an integer and n > 2.

(a) Find the marginal PDF of X and its mean.

(b) Find the conditional PDF of Y given X = x.

(c) Deduce the conditional mean and the conditional variance of Y given X = x.

(d) Find the mean and variance of Y .

(e) Find the covariance of X and Y by conditioning on X.                         [18 marks]

8 The joint PDF of X and Y is given by

(a) Find c and the correlation coefficient of X and Y .

(b) Find the best least square estimator of Y based on X.           [5 marks]

9 Let Sn denote the sum of n independent Poisson random variables with parameter 1.

(a) Show that Sn is a Poisson random variable and give its parameter.

(b) Write down an expression for P(Sn ≤ n) in terms of n.

(c) Use the Central Limit Theorem to show that

                 [8 marks]