ST3236: Stochastic Process Tutorial 2 TA: Mar Choong Hock Email: g0301492@nus.edu.sg Exercises: 3

Question 1 Four nickels and six dimes are tossed, and the total number N of heads is observed. If N = 4, what is the conditional probability that exactly two of the nickels were heads?

Question 1 A is the event that N = 4. B is the event that exactly two of the nickels were heads.  A B

Question 1 Let X1 be the number of heads of the nickels and X2 the number of heads of the dimes. Then, N = X1 + X2 P(X1 = 2 | N = 4) = P(X1 = 2,X1 + X2 = 4) / P(X1 + X2 = 4) = P(X1 = 2, X2 = 2) / P(X1 + X2 = 4) = P(X1 = 2)P(X2 = 2) / P(X1 + X2 = 4) (assuming the probabilities of heads are 0.5)

Question 2 A dice is rolled and the number N on the uppermost face is recorded. From a jar containing 10 tags numbered 1, 2, ..., 10 we then select N tags at random without replacement. Let X be the smallest number on the drawn tags. Determine P(X = 2).

Question 2

Question 2

Question 3 Let X be a Poisson random variable with parameter, . Find the conditional mean of X, given that X is odd.

Question 3 (define that 0 is even) Let pk = P(X = k).

Question 3

Question 3 Note: coth (.) denotes hyperbolic cotangent.

Question 3 - Optional An alternative solution without using p.g.f.:

Question 3 - Optional Note: cosh (.) denotes hyperbolic cosine.

Question 4 Suppose that N has density function: P(N =n) = (1-p) n-1p for n = 1, 2,… where p in (0, 1) is a parameter. This defines the geometric distribution with parameter p.

Question 4a Show that: G(s) = sp/[1 - s(1 - p)] for s < 1/(1 - p).

Question 4b Show that: E(N) = 1/p

Question 4c Show that: var(N) = (1 - p)/p2

Question 4d Show that: P(N is even) = (1 - p)/(2 - p).