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# The number of (large) inclusions in cast iron follows a

ISBN: 9781118539712 55

## Solution for problem 111E Chapter 4.7

Applied Statistics and Probability for Engineers | 6th Edition

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Problem 111E

The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Approximate the following probabilities:

(a) Determine the mean and standard deviation of the number of inclusions in a cubic centimeter (cc).

(b) Approximate the probability that fewer than 2600 inclusions occur in a cc.

(c) Approximate the probability that more than 2400 inclusions occur in a cc.

(d) Determine the mean number of inclusions per cubic millimeter such that the probability is approximately 0.9 that 500 or fewer inclusions occur in a cc.

Step-by-Step Solution:

Step 1 of 4</p>

(a)

The number of inclusions in cast iron follows a Poisson distribution with a mean of

We are asked to find the mean and standard deviation of the number of inclusions in a cubic centimeter

Let  denote the number of inclusions in a cubic centimeter . Then,  has a Poisson distribution with

……(1)

If  is a Poisson random variable over an interval of length  with parameter , then

Using equation (1), we can write the mean and variance,

Hence standard deviation,

Hence the mean and standard deviation of the number of inclusions in a cubic centimeter  is  respectively.

Step 2 of 4</p>

(b)

We are asked to approximate the probability that fewer than  inclusion occurs in a cc.

We need to find the probability

If  is a Poisson random variable with  and

Is approximately a standard normal variable.

The same continuity correction of used for the binomial distribution can also be applied.

The approximation is good for

The computational difficulty is clear. Hence the probability can be approximated after continuity correction of  as

From the z table, the area to the left of  is .

Hence the approximate probability that fewer than  inclusion occur in a cc is

Step 3 of 4

Step 4 of 4

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