# Random Variable and Probability Distribution Function

**Topics:**Random variable, Normal distribution, Probability density function

**Pages:**19 (5518 words)

**Published:**December 12, 2012

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 |0.22 |0.17 |0.13 |0.05 |0.02 |

Find the mean and standard deviation of the distribution.

5. The tables below are the probability distribution functions of number of sick leave taken in a month by male and female employees in a large company.

X: number of sick leave taken by male

|x |0 |1 |2 |3 |4 |5 | |P(X = x) |0.3 |0.29 |0.24 |0.12 |0.03 |0.02 |

Y: number of sick leave taken by female

|y |0 |1 |2 |3 |4 |5 | |p(Y = y) |? |0.32 |0.34 |0.06 |0.04 |0.03 |

What are the expected numbers of sick leave taken in a month by male and female employees respectively? What are the standard deviations of numbers of sick leave taken in a month by male and female employees respectively?

6.At the time of purchase of its product, a company offers a contract to its customer that guarantees the replacement of the product if it malfunctions within a one-year period. The company receives $10 from the customer for each contract. The probability that a unit of the product malfunctions within one year of purchase is 0.06 and the replacement cost is $160. What is the company’s expected gain (or loss) per contract?

7.A state lottery is to be conducted in which 10,000 tickets are to be sold for $1 each. Six winning tickets are to be randomly selected: one grand-prize winner of $5,000, one second-prize winner of $2,000, one third-prize winner of $1,000, and three other winners of $500 each. a.Compute the expected gain of playing this game.

b. Would you play this game? Why?

8.Let [pic] be a random variable having the following distribution:

|x |-1 |0...

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