Question

• / 5 points

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the world bank records the prevalence of hiv in countries around the world. according to their data, \prevalence of hiv refers to the percentage of people ages 15 to 49 who are infected with hiv.\ in one country, the prevalence of hiv is 18.5%. let
x = the number of people you test until you find a person infected with hiv
part (a)
part (b)
part (c)
part (d)
find the mean and standard deviation of the distribution of x. (round your answers to four decimal places.)
(i) mean
(ii) standard deviation

Explanation:

Step1: Identify distribution type

$X$ follows a geometric distribution, where the probability of success (finding an infected person) is $p = 0.185$.

Step2: Calculate the mean

For a geometric distribution, the mean is $\mu = \frac{1}{p}$.
$\mu = \frac{1}{0.185}$

Step3: Compute the mean value

$\mu \approx 5.4054$

Step4: Calculate the standard deviation

For a geometric distribution, the standard deviation is $\sigma = \frac{\sqrt{1-p}}{p}$.
$\sigma = \frac{\sqrt{1-0.185}}{0.185}$

Step5: Compute the standard deviation value

First, $1-0.185 = 0.815$, so $\sqrt{0.815} \approx 0.9028$. Then $\sigma = \frac{0.9028}{0.185} \approx 4.8800$

Answer:

(i) mean: $\boldsymbol{5.4054}$
(ii) standard deviation: $\boldsymbol{4.8800}$