Question

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

x	1	2	3
f	6	8	6

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 2, a value of at most 2, and a value greater than 2.

c. Find P(X = 2), P(X≤ 2), and P(X > 2). Interpret your results.

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

Short answer

Part a.

Random Variable (X)	Probability
1	6/20 = 0.3
2	8/20 = 0.4
3	6/20 = 0.3

Part b.

On the value of 2: X = 2

A value of at most 2 : X ≤ 2

A value greater than 2: X > 2

Part c.

$$\begin{gathered} P(X=2)=0.4 \\ P(X\le 2)=0.3+0.4 \\ =0.7 \\ P(X>2)=0.3 \end{gathered}$$

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	1	2	3
f	6	8	9

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 ,$N=6+8+6=20$. As a result, the probability of random variable X is as follows:

Random Variable	Probability
1	$\frac{6}{20}=0.3$
2	$\frac{8}{20}=0.4$
3	$\frac{6}{20}=0.3$

Part (b) Step 1. Solution 

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

X = 2

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

X ≤ 2

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

X > 2

Part (c) Step 1. Solution

From the table, we get :

$$\begin{gathered} P(X=2)=0.4 \\ P(X\le 2)=0.3+0.4 \\ =0.7 \\ P(X>2)=0.3 \end{gathered}$$

Part (d) Step 1. Solution

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