Question

According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one. Out of a randomly chosen group of 600 U.S. women determine the following.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution ofX. X ~ _____(_____,_____)

d. How many are expected to suffer from anorexia?

e. Find the probability that no one suffers from anorexia.

f. Find the probability that more than four suffer from anorexia.

Short answer

a. The random variable of X is the number of women who suffer from anorexia.

b. The values of X are $0,1,2,3,............,600$

c. The distribution of X is $X~P\left(3\right)$

d. The expected number of women who suffer from anorexia is $3$

e. The probability that no one suffers from anorexia is $0.0498$

f. The probability that more than four suffer from anorexia is$0.1847$

Step-by-step solution

Content Introduction

In a large population, the Poisson distribution is used to characterize the distribution of unusual events.

Explanation (part a)

We are given,

The average number of women who suffer from anorexia is one and randomly we have chosen group of 600 women.

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case X is the number of women who suffer from anorexia.

Explanation (part b)

The list of values that we have for X are$0,1,2,3,.............,600.$

Explanation (part c)

Here, the random variable X refers to the number of trials. Each trial is independent of others and has similar probability of success. This implies that random variable X follows Poisson Distribution.

The Poisson distribution is shown as $X~P(x,\mu )$where x is the successes occurs in poisson distribution and $\mu$is the mean successes.

Thus, the distribution of X is$X~P\left(3\right)$

Explanation (part d)

We are given, the average number who suffer from anorexia is one and out of a randomly chosen group, 600 U.S. women suffer from the disease.

The expected number of women who suffer from anorexia is:

$$\begin{gathered} X=1\times 3 \\ X=3 \end{gathered}$$

Explanation (part e)

The probability that no one suffers from anorexia is:

$$\begin{gathered} P_n(X=0)=\frac{3^0}{0!}{e}^{-3} \\ {P}_n(X=0)=0.0498 \end{gathered}$$

Explanation (part f)

The probability that more than four suffer from anorexia is:

$$\begin{gathered} P_n(\xi \le 4)=P\left(0\right)+P\left(1\right)+P\left(2\right)+P\left(3\right)+P\left(4\right) \\ {P}_n(\xi \le 4)=0.00335+0.01907+0.05436+0.01327 \\ {P}_n(\xi \le 4)=0.1847 \end{gathered}$$