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.

Answer

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