Estimating the Mean for a Batch of 50 Items Using the Poisson Distribution

[naspek]

A machine on average produces 4 defective items out of a batch of 100 items.
Find the probability that a batch of 50 items has 3 defective items in it using the Poisson probability distribution.

the problem is..
i just want to know the mean or average value for batch of 50 items..
i got mean = 2
because for 100 items, the mean is 4..
...for 50 items, the mean would be 2..
correct?

 

[Mark44]

Here's how I would start it. Let the random variable X be the number of defective items in a batch of 50. P(X = 1) = 0.04.

Assuming that X is Poisson with mean lambda,
$$P(X = k)~= ~\frac{\lambda^k~e^{-k}}{k!}$$
We also have P(X = 1) = 0.04, so using the equation above, I get
$$\frac{\lambda^1~e^{-\lambda}}{1}~=~0.04$$

This isn't an equation that you can solve analytically, but you can use estimation techniques to get approximate values for lambda. In about a minute I got a value for lambda of about 4.8. The better you estimate for lambda is, the better your calculation for P(X = 3) will be.