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Homework 5 Due: Tuesday, July 17th (before class begins)

1. A shipment of 30 similar laptop computers to a retail outlet contains 10 that are defective. A school makes a random purchase of 5 of these computers. Let X denotes the number of defective

computers.

a. Find the probability distribution of X. b. Find mean and variance of X. c. Find 𝑬(𝟒𝑿 + 𝟏) 𝒂𝒏𝒅 𝑽𝒂𝒓(𝟑𝑿 − 𝟏𝟎).

2. An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let of storage space purchased by the next customer to

buy a freezer. Suppose that X has pmf

a. If the price of a freezer having capacity X cubic feet is 25X - 8.5, what is the expected price paid by the next customer to buy a freezer?

b. What is the variance of the price 25X -8.5 paid by the next customer? c. Suppose that although the rated capacity of a freezer is X, the actual capacity is

𝒉(𝑿) = 𝑿 − 𝟎. 𝟎𝟏 𝑿𝟐. What is the expected actual capacity of the freezer purchased by the next customer?

3. A random variable X has the following pmf:

𝒙 1 3 4 6 10 12 16 𝒑(𝒙) 0.1 0.15 0.2 0.25 0.15 0.1 0.05

a. Find the cumulative distribution function of X, F(x). b. Sketch a graph of F(x) c. compute 𝑷(𝟑 ≤ 𝑿 ≤ 𝟔) 𝒂𝒏𝒅 𝑷(𝑿 > 𝟒)

4. A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages and consider a sample of 12 incoming calls. What

is the probability that

a. at most 5 of the calls involve a fax message? b. at least 3 of the calls involve a fax message? c. What is the expected number of calls among the 12 that involve a fax message? d. What is the standard deviation of the number among the 12 calls that involve a fax

message?

e. What is the probability that the number of calls among the 12 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?

5. Calls received by a car rescue service occur independently and at a constant average rate of 3 per minute.

a. Find the probability that, in a randomly chosen period of 1 minute, the number of calls received by the service is

(I ) at most 3

(II) at least 3

(III) between 2 and 5 (inclusive)

b. Find the probability that, in a randomly chosen period of 2 minute, the number of calls received by the service is at most 5.

𝒙 13.5 15.9 19.1 𝒑(𝒙) 0.2 0.5 0.3