The mean of the probability distribution function given by the following table will be:

x	1	2	3	4	5
P(x)	0.2	0.35	0.25	0.15	0.05

 

Concept:

Expected Value (or mean) of a Discrete Random Variable:
For a discrete random variable, the expected value usually denoted as μ or E(X), is calculated using

μ = E(X) = ∑ x_iP(x_i)

1. The formula means that we multiply each value x, in the support by its respective probability P(x) and then add them all together.

2. It can be seen as an average value but weighted by the likelihood of the value.

Calculation:

We know that, mean

μ = E(X) = ∑ x_iP(x_i)

⇒ E(X) = 1 × 0.2 + 2 × 0.35 + 3 × 0.25 + 4 × 0.15 + 5 × 0.05

⇒ E(X) = 0.2 + 0.7 + 0.75 + 0.6 + 0.25

⇒ E(X) = 2.5

Additional Information
The variance of a discrete random variable is given by:

Var(X) = σ²(X) = E(X²) - [E(X)]²

The formula means that first, we sum the square of each value times its probability then subtract the square of the mean.