Random Variable and Probability Distribution Function

Exercise
Chapter 3 Probability Distributions

1. Based on recent records, the manager of a car painting center has determined the following probability distribution for the number of customers per day. Suppose the center has the capacity to serve two customers per day.

|x |P(X = x) |
|0 |0.05 |
|1 |0.20 |
|2 |0.30 |
|3 |0.25 |
|4 |0.15 |
|5 |0.05 |

a. What is the probability that one or more customers will be turned away on a given day? b. What is the probability that the center’s capacity will not be fully utilized on a day? c. At least by how many, the capacity must be increased so the probability of turning a customer away is no more than 0.1?

2. The following is the probability distribution function of the number of complaints a customer manager has to handle in half an hour.

Suppose he can handle at most 3 complaints in half an hour. a. What is k? b. What is the probability there are less than 2 complaints in half an hour? c. What is the probability there are less than 2 complaints in an hour?

3. A random variable [pic] can be assumed to have five values: 0, 1, 2, 3, and 4. A portion of the probability distribution is shown here:

|x |0 |1 |2 |3 |4 |
|P(X = x) |0.1 |0.3 |0.3 |a |0.1 |

a. Find a b. Find [pic], E(X2), [pic] and the standard deviation of [pic].

4. The probabilities that a building inspector will observe 0, 1, 2, 3, 4, or 5 (X) violations of the building code in a home built in a large development are given in the following table:
|x |0 |1 |2 |3 |4 |5 |
|P(X = x) |0.41