Mathematical expectation
The concept of mathematical expectation is of big importance in the statistical work. The mathematical expectation also known as the expected value of a random variable is the weighted arithmetic mean of the variable, the weights are used to find the mathematical expectation are all the respective probabilities of the value that the variable can possibly suppose.
If X denotes a discrete random variable that can assume the values X1, X2, X3,......, Xk, with their respective probability P1, P2, P3,........., Pk where p1 + p2 + p3 +.....+ Pk = 1 is the mathematical expectation of X represented by E (X) and is defined as:
E (X) = p1 X1 + p2 X2 + p3 X3 + .......+ Pk Xk
Example: - a petrol pump proprietor sells on an average $ 80.000 worth of petrol on rainy days and an average of $ 95,000 on clear days. The Statistics from the meteorological department represent that the probability is 0.76 for the clear weather and 0.24 for rainy weather on coming Monday. Compute the expected value of petrol sale on coming Monday.
Solution: - x1 = 80,000 P1 = 0.24
X2 = 95,000 P2 = 0.76
Σ (X) = P1 X1 + P2 X2 + ................
= 0.24 (80.000) + 0.76 (95,000)
= 19,200 + 72,200 = $ 91,400
Therefore, the expected value of petrol sale on coming Monday is $ 91,400.
Example: a firm plans to bid $300 per tonne for a contract to supply 1000 tonnes of a metal. It has two competitors A & B and it suppose that the probability that A will bid less than $300 per tonne is 0.3 and that B will be less than $ 300 per tonne is 0.7. If the minimum bidder gets all the business and if the firms bid separately, then compute the expected value of the contract to the firm?
Solution: there are two competitors A & B and the lowest bidder get the contract.
Value of plan = 300 × 1000 = $ 300000
Contractor A: probability that bid is less than $ 300 per tonne = 0.3
Probability that bid is $ 300 or more = 0.7
Contractor B: probability that bid is less than $300 per tonne = 0.7
The Probability that bid is $300 or more per tonne = 0.3
1. If both bids are less than $300
Probability is 0.3 × 0.7 = 0.21
Therefore plan value is 300000 × 0.21 = 63000
2. If only A bids less than 300 & B bids more than 300
Probability is 0.3 × 0.3 = 0.09
Thus, plan value is 300000 × 0.09 = 27000
3. B bids less than 300 while A bids more than 300
Probability is 0.7 × 0.7 = 0.49
Thus plan value is 300000 × 0.49 = 147000
Thus, value of plan is $ = 63000 + 27000 + 147000
= $ 237000.
And hence, the expected value of plan is $ 237000.