Question

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

y	0	1	4	6
f	18	14	8	10

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

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

b. Use random-variable notation to describe the events that Y takes on the value 3, a value less than 3, and a value of at least 3.

c. Find P(Y = 3), P(Y < 3), and P(Y 3). Interpret your results.

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

Short answer

Part a.

Random Variable	Probability
0	0.36
1	0.28
4	0.16
6	0.20

Part b.

On the value of 3: Y = 3

A value less than 3 : Y < 3

A value of at least 3 : Y ≥ 3

Part c.

$$\begin{gathered} P(Y=3)=0 \\ P(Y<3)=0.64 \\ P(Y\ge 3)=0.36 \end{gathered}$$

Part d.

Step-by-step solution

Part (a) Step 1. Given information

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

y	0	1	4	6
f	18	14	8	10

Let Y signify the value of the variable y 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, $N=18+14+8+10=50$. As a result, the probability of random variable Y is as follows:

Random Variable	Probability
0	18/50 = 0.36
1	14/50 = 0.28
4	8/50 = 0.16
6	10/50 = 0.20

Part (b) Step 1. Solution 

Event where Y takes the value 3, can also be written as:

Y = 3

Event where Y takes a value less than 3, can also be written as :

Y < 3

Event where Y takes a value of at least 3, can also be written as :

Y ≥ 3

Part (c) Step 1. Solution 

From the table,

$$\begin{gathered} P(Y=3)=0 \\ P(Y<3)=0.36+0.28=0.64 \\ P(Y\ge 3)=0.16+0.20=0.36 \end{gathered}$$

Part (d) Step 1. Solution 

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