# Poisson Probability Distribution

1. Feb 27, 2010

### exitwound

1. The problem statement, all variables and given/known data

Suppose that .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers.

a.)What are the expected value and standard deviation of the number of computers in the sample that have the defect?
b.) What is the (approximate) probability that more than 10 sampled computers have the defect?
c.) What is the (approximate) probability that no sampled computers have the defect?

2. Relevant equations

$$p(x;\lambda )=\frac{e^{-\lambda}\lambda^x}{x!}$$

3. The attempt at a solution

a.) E(X) of a poisson distribution is $\lambda$ which is np which is (10,000)(.001)=10.

V(X) is also $\lambda$ or 10.

The standard deviation is the squareroot of V(X) or sqrt(10)?

b.) To do this, would I take the sum of the probability of 10 machines having a failure to infinity cpus failing? (i.e. p(10;10) + p(11;10) + ... p(inf,10))

$$p(x\geq 10;10)=\sum_{x=10}^{\infty} {\frac{e^{-10}10^x}{x!}}$$

c.) To find this, would I use p(0;10)? If so:

$$p(0;10)={\frac{e^{-10}10^0}{0!}}$$

$$p(0;10)={\frac{e^{-10}(1)}{1}}=e^{-10}$$

2. Feb 27, 2010

### snipez90

Looks fine except in the second part, more than 10 probably means strict inequality and since poisson is discrete, you probably need to index the sum starting at x = 11.

3. Feb 27, 2010

### exitwound

Ah yes. Thank you :)