Question

Mental health in America. In the Mental Health Surveillance Study, clinicians interviewed a very large random sample of American adults so as to diagnose the presence of mental disorders in the prior 12 months. There are many types of mental disorders, and a person can suffer from more than one. Let X represent the number of mental disorders a person has experienced in the past year. Based on this study, we can assign the following probability distribution to X in the American adult population:

X 0 1 2 3 4 or more
P(X) 0.775 0.149 0.041 0.022 0.013
(a) Explain why this is a valid probability distribution. Is the sample space discrete orcontinuous?
(b) Find P(X <2) and P(1≤X≤3).
(c) Obtain the probability that a randomly selected American adult suffers from
(i) at least one mental disorder
(ii) either one or two mental disorders

Answer 1

a) It is a valid distribution because the sum of all possible probabilities is 1. The sample space is discrete, because the number of diseases is a countable number.

b) P(X <2) = 0.924, P(1≤X≤3) = 0.212

c - i) 0.225

c - ii) 0.19

Explanation:

(a) Explain why this is a valid probability distribution. Is the sample space discrete or continuous?

0.775 + 0.149 + 0.041 + 0.022 + 0.013 = 1

Since the sum of all possible probabilities is 1, this is a valid probability distribution.

Since the possible outcomes are 0, 1, 2, 3 or 4 diseases, that is, countable numbers, the sample space is discrete.

(b) Find P(X <2) and P(1≤X≤3).

`P(X < 2) = P(X = 0) + P(X = 1) = 0.775 + 0.149 = 0.924`

`P(1 \geq X \geq 3) = P(X = 1) + P(X = 2) + P(X = 3) = 0.149 + 0.041 + 0.022 = 0.212`

(c) Obtain the probability that a randomly selected American adult suffers from

(i) at least one mental disorder

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.775 = 0.225`

(ii) either one or two mental disorders

`P(X = 1) + P(X = 2) = 0.149 + 0.041 = 0.19`