Question

A light bulb produced by a factory is known to be 600 hours or more (undamaged) with a probability of 0.95. A sample of 10 lamps was purchased. Make a probability distribution table for the number of undamaged lights X. Calculate the expectation / mean µ and the standard deviation of X.

Answer 1

Given n = 10 the number of purchased lamps and the probabilities of undamaged lamps is p, where p = 0.95. Let X be a random variable, which describes the number x of undamaged lamps purchased. The random variable X corresponds to the Binomial distribution with parameters x<=n and p (see Wikipedia for example). Thus

$Pr[X=x] = \binom{n}{x}p^x (1-p)^{n-x}$ . The mean $\mu$ and the standard deviation $\sigma$ of X given by

$\mu = n*p = 9.5$ ;

$\sigma =\sqrt{n*p*(1-p)}=\\\sqrt{10*0.95*(1-0.95)} \approx 0.689$ .

The table is depicted below