PSTAT160A - Introduction to Stochastic Processes Brunick
PSTAT160A - Practice Problems for the Midterm
You can expect problems like the following in the midterm. It is unlikely that I will have an
opportunity to write up solutions to these problems.
1. A coin is randomly selected from a group of ten coins where the ith coin has probability i/10
of landing heads with i= 1,2,...,10. This coin is then flipped until a heads is obtained. Let
the random variable Ndenote the number flips that are required. Compute the probability
mass function and expected value of N. Don’t expect everything to simplify.
2. There are three types of light bulbs on the market: produced by company A,B, or C. It is
known that 20% of the bulbs on the market are produced by company A, 30% by Band 50%
by C. It is also known that 3% of the bulbs produced by Aare defective, 5% of the bulbs
produced by Bare defective and 2% of the bulbs produced by Care defective. Once the
bulbs are on the market, a customer cannot distinguish which company produced it. Assume
that you buy a light bulb and it is defective. What is the probability that it was produced
by company B?
3. Let Xbe a geometric random variable with parameter p1, and let Ybe a geometric random
varaible with parameter p2. Assume that Xand Yare independent and compute P(X≤Y).
4. A fair coin is tossed repeatedly and the record of the outcomes is kept. Tossing stops the
moment the total number of heads obtained so far exceeds the total number of tails by 3.
For example, a possible sequence of tosses could look like HHTTTHTHHTHH. What is the
probability that the length of such a sequence is at most 10? You can compute the answer
exactly.
5. Let (Xn)n≥0be a simple symmetric random walk and denote by (Nn)n≥0its running mini-
mum, which means Nn= min(X0, X1,...Xn).
(a) Compute P(X10 = 2 and X18 = 6).
(b) Compute P(X12 = 2 and N12 ≥ −3).
(c) Compute P(X8= 2 and N8=−1).
1/1