A variable x of a finite population has the following frequency distribution :

x	5	7	8
f	25	40	60

Suppose a member is selected at random from the population and let X denote the value of the variable x for the member obtained.

a. Determine the probability distribution of the random variable X.

b. Use random-variable notation to describe the events that X takes on the value 6, a value of at most 6, and a value greater than 6.

c. Find P(X = 6), P(X < 6), and P(X > 6). Interpret your results.

d. Construct a probability histogram for the random variable X.

The frequency distribution of a variable x in a finite population is as follows :

Let X signify the value of the variable x for the member acquired if a member is chosen at random from the population.

The experiment is conducted out assuming that the random variable (X=x) occurs n times out of a total of N times. As a result of the $\frac{f}{N}$rule:

$P(X=x)=\frac{n}{N}$

Here, $N=25+40+60=125$. Hence the probability of random variable X is:

Event where X takes the value 2, can also be written as:

X = 6

Event where X takes a value of at most 2, can also be written as :

X ≤ 6

Event where X takes a value greater than 2, can also be written as :

X > 6

From the table,

$$\begin{gathered} P(X=6)=0 \\ P(X\le 6)=0.2 \\ P(X>6)=0.32+0.48 \\ =0.8 \end{gathered}$$

The probability histogram for the random variable X is as follows: