1. ST3236: Stochastic Process Tutorial 2
TA: Mar Choong Hock
Exercises: 3

2. 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?

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

4. 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)

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).

6. Question 2

7. Question 2

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

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

10. Question 3

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

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

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

14. 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.

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

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

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

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