Prove that the Px(x) is probability mass function
Px (x)=( I\14 x^ 2 if x £[1,2,3,4])
0. Otherwise
To prove that Px(x) is a probability mass function, we need to show that it satisfies two key properties of a probability mass function:
1. For any value of x, 0 ≤ Px(x) ≤ 1
2. The sum of Px(x) over all possible values of x is equal to 1
Given the function Px(x):
Px(x) = { 1/4 * x^2 if x ∈ {1, 2, 3, 4}
0 otherwise
Let's check the first property:
1. For x = 1, 2, 3, 4:
- Px(1) = 1/4 * 1^2 = 1/4 ≤ 1
- Px(2) = 1/4 * 2^2 = 1 ≤ 1
- Px(3) = 1/4 * 3^2 = 9/4 ≤ 1
- Px(4) = 1/4 * 4^2 = 4 ≤ 1
For x not in {1, 2, 3, 4}, Px(x) = 0 which clearly satisfies 0 ≤ Px(x) ≤ 1 for all values of x.
Next, let's check the second property:
2. Sum of Px(x) over all possible values of x is equal to 1:
Px(1) + Px(2) + Px(3) + Px(4)
= 1/4 * 1^2 + 1/4 * 2^2 + 1/4 * 3^2 + 1/4 * 4^2
= 1/4 + 1 + 9/4 + 4
= 1
Since the sum of Px(x) over all possible values of x equals 1, Px(x) satisfies the second property of being a probability mass function.
Therefore, Px(x) is indeed a probability mass function.