Question

A telemarketer knows that, on average, he is able to make three sales in a 45-minute period. Suppose the number of sales he can make in a given time period is Poisson distributed.

a) What is the probability that he makes two sales in a 30 -minute period?

b )What is the probability that he makes at least two sales in a 30-minute period?

c) What is the probability that he makes only one sale in an hour-long period?

Please provide good explanation.

Answer 1

Let X denote the random variable representing number of sales by the telemarketer in a 30-minute period.

For parts a) and b) consider that fact that the telemarketer, on average, is able to make 3 sales in a 45-minute period. Thus, in a 30-minute period he, on average, is able to make 3*(30/45) = 2 sales.

=> E(X) = 2 ................(1)

Now, we are given that number of sales the telemarketer can make in a given time period is Poisson distributed.

=> X ~ Poisson(λ) for some parameter λ.

=> E(X) = λ ...................(2)

From equations (1) and (2), we get:

λ = 2

Thus, we get: X ~ Poisson(λ = 2) and the probability mass function of X is given by:

a)

The probability that he makes two sales in a 30-minute period is given by:

b)

The probability that he makes at least two sales in a 30-minute period is given by:

c)

Let Y denote the random variable representing number of sales by the telemarketer in a 1-hour period.

Now, consider that fact that the telemarketer, on average, is able to make 2 sales in a 30-minute period. Thus, in a 1-hour period (or a 60-minute period) he, on average, is able to make 2*2 = 4 sales.

=> E(Y) = 4 ................(3)

Now, we are given that number of sales the telemarketer can make in a given time period is Poisson distributed.

=> Y ~ Poisson(μ) for some parameter μ.

=> E(Y) = μ ...................(4)

From equations (3) and (4), we get:

μ = 4

Thus, we get: Y ~ Poisson(μ = 4) and the probability mass function of Y is given by:

Thus, the probability that he makes only one sale in an hour-long period is given by:

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