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Math/Stat 318 Midterm 2 Practice

1. Assume by previous research we know every customer that visits a website has a 3% chance of making a purchase.

(a) What is the probability that the 2nd purchase is made by the 10th customer to visit the website?

(b) If 20 customers visit the website what is the probability that exactly 1 purchase is made?

(c) Each day, a big purchase (over $100) is made with probability 0.01. What is the probability that the first big purchase happens on the 6th day the website is live?

2. Find P(X = 1), E(X), and V ar(X) if the mgf of X is:

MX(t) = 0.35 + 0.65e^t

3. Let X be a continuous random variable whose probability density function is:

f(x) = 5e^x , x ∈ (0, c)

What is the value of the constant c, so that makes f(x) a valid probability density function?

4. Suppose customers arrive at a shop on the average of 30 per hour in accordance with a Poisson process.

(a) What is the probability that the shopkeeper will have to wait longer than 9.488 minutes for the first two customers to arrive?

(b) On average, how long does the shopkeeper have to wait for the first two customers to arrive?

5. Let X be a chi-square random variable with 20 degrees of freedom.

(a) What is the upper 10th percentile for the distribution of X?

(b) What is the probability that X is greater than 37.57?

6. Let Z ~ U(30, 40). Use the moment generating function of Z to derive the mean.