Question

use the fact that the mean of a geometric distribution is \\( \mu = \frac{1}{p} \\) and the variance is \\( \sigma^2 = \frac{q}{p^2} \\).

a daily number lottery chooses three balls numbered 0 to 9. the probability of winning the lottery is \\( \frac{1}{1000} \\). let \\( x \\) be the number of times you play the lottery before winning the first time.
(a) find the mean, variance, and standard deviation. (b) how many times would you expect to have to play the lottery before winning? it costs $1 to play and winners are paid $400. would you expect to make or lose money playing this lottery? explain.

(a) the mean is 1000. (type an integer or a decimal.)
the variance is 999000. (type an integer or a decimal.)
the standard deviation is \\( \square \\).
(round to one decimal place as needed.)

Explanation:

Step1: Recall standard deviation formula

The standard deviation \(\sigma\) is the square root of the variance \(\sigma^2\). So we use \(\sigma = \sqrt{\sigma^2}\).

Step2: Substitute variance value

We know the variance \(\sigma^2 = 999000\). So we calculate \(\sigma=\sqrt{999000}\).
Calculating \(\sqrt{999000}\approx999.5\) (rounded to one decimal place).

Answer:

999.5