Question

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.

Short answer

Part a.

Random Variable	Probability
5	0.2
7	0.32
8	0.48

Part b.

On the value of 6: X = 6

A value of at most 6: X ≤ 6

A value greater than 6: X > 6

Part c.

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

Part d.

Step-by-step solution

Part (a) Step 1. Given information

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

x	5	7	8
f	25	40	60

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

Part (a) Step 2. Formula Used

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:

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

Part (a) Step 3. Solution

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

Random Variable	Probability
5	25/125 = 0.2
7	40/125 = 0.32
8	60/125 = 0.48

Part (b) Step 1. Solution 

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

Part (c) Step 1. Solution

From the table,

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

Part (d) Step 1. Solution

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