# Probability Distribution Mean (Expectation), Variance :: Problems

 Problem Back to Problems Page
The monthly demand for radios is known to have the following probability distribution
 Demand Probability 1 2 3 4 5 6 0.1 0.15 0.2 0.25 0.2 0.1
Determine the expected demand for radios. Also find the variance. If the cost (Rs. C) of producing (n) radios is given by C = 1000 + 200n, determine the expected cost.

 [Expectation Cost: 1,720]

 Solution

Let "x" represent the demand for radios.

The demand for radios can be 1, 2, 3, ... , 6
⇒ "x" can carry the values 1, 2, 3, ... , 6
⇒ "x" is finite i.e. discrete

Therefore, the given probability distribution would be a discrete probability distribution of a random variable "X"

For a discrete probability distribution of a random variable "X", Σ p = 1

Calculations for finding the mean/expectation and variance of the distribution

x P (X = x) px
[x × P (X = x)]
x2 px2
[x2 × P (X = x)]
1 0.10 0.10 1 0.10
2 0.15 0.30 4 0.60
3 0.20 0.60 9 1.80
4 0.25 1.00 16 4.00
5 0.20 1.00 25 5.00
6 0.10 0.60 36 3.60
Total 1.00 3.60 15.10

⇒ Mean/Expectation of the distribution
 ⇒ E (x) (Or) x = Σ px = 3.60

⇒ Variance of the distribution
 ⇒ var (x) = E (x2) − (E(x))2 ⇒ var (x) = Σ px2 − (Σ px)2 = 15.10 − (3.60)2 = 15.10 − 12.96 = 2.14

⇒ Standard Deviation of the distribution
 ⇒ SD (x) = + √ Var (x) = + √ 2.14 = + 1.463

The expected cost
⇒ Expected cost of producing radios to meet the expected demand.

Expected Cost is given by the relation 1000 + 200n where "n" represents no. of radios

Therefore, for the expected demand, n = 3.60
 ⇒ Expected Cost = Cost of producing 3.60 radios ⇒ C = 1000 + 200 × (3.60) = 1000 + 720 = 1,720

## Alternative for finding expected cost

The costs of radios based on the monthly demand would be:
Demand Probability Calculations for cost Cost
1 0.10 1000 + 200 (1) 1,200
2 0.15 1000 + 200 (2) 1,400
3 0.20 1000 + 200 (3) 1,600
4 0.25 1000 + 200 (4) 1,800
5 0.20 1000 + 200 (5) 2,000
6 0.10 1000 + 200 (6) 2,200
The monthly costs of radios demanded is also a probability distribution
 Cost of Radios Demanded Probability 1,200 1,400 1,600 1,800 2,000 2,200 0.10 0.15 0.20 0.25 0.20 0.10

Calculations for finding the mean/expectation and variance of the distribution

x P (X = x) px
[x × P (X = x)]
1,200 0.10 120
1,400 0.15 210
1,600 0.20 320
1,800 0.25 450
2,000 0.20 400
2,200 0.10 220
Total 1.00 1,720

Expected Cost
⇒ Mean/Expectation of the distribution
 ⇒ E (x) (Or) x = Σ px = 1,720

 Credit : Vijayalakshmi Desu