Question

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.”[1] In South Africa, the prevalence of HIV is 17.3%. Let X = the number of people you test until you find a person infected with HIV.

a. Sketch a graph of the distribution of the discrete random variable X.

b. What is the probability that you must test 30 people to find one with HIV?

c. What is the probability that you must ask ten people?

d. Find the

(i) mean and

(ii) standard deviation of the distribution of X.

Short answer

a. The graphic presentation is

b. The probability that we must test 30 people to find one with HIV is $0.0007$

c. The probability that we must ask ten people $0.0313$

d mean is$5.7803$and standard deviation is$5.2566$

Step-by-step solution

Content Introduction

In a Bernoulli trial, the likelihood of the number of successive failures before a success is obtained is represented by a geometric distribution, which is a sort of discrete probability distribution. A Bernoulli trial is a test that can only have one of two outcomes: success or failure.

Explanation (part a)

Variable X has a geometric distribution with the probability of success $0.173$.

Therefore, the graphic presentation is as follow:

Explanation (part b)

The probability that we need to test 30 people to find one with HIV is calculated as:

$$\begin{gathered} P(X=30)={(1-0.173)}^{x-1}\times 0.173 \\ P(X=30)={(0.827)}^{29}(0.173) \\ P(X=30)=0.0007 \end{gathered}$$

Explanation (part c)

The probability that we must ask ten people is as follow:

$$\begin{gathered} P(X=10)={(1-0.173)}^{n-1}(0.173) \\ P(X=10)={(0.827)}^9(0.173) \\ P(X=10)=0.0313 \end{gathered}$$

Explanation (part d)

The mean of the variable with geometric distribution with parameter $p=0.173$

Mean$=\frac{1}{p}$

$$\begin{gathered} Mean=\frac{1}{0.173} \\ Mean=5.7803 \end{gathered}$$

Standard deviation is $\delta =\sqrt{\frac{1-p}{p^2}}$where$p=0.173$

$$\begin{gathered} \delta =\sqrt{\frac{1-p}{p^2}} \\ \delta =\sqrt{\frac{1-0.173}{{(0.173)}^2}} \\ \delta =5.2566 \end{gathered}$$