Page 143 - DMTH404_STATISTICS

Unit 10: Approximate Expressions for Expectations and Variance

Notes
5 4 3 2 1 252
+ 8 ´ + 9 ´ + 10 ´ + 11 ´ + 12 ´ = = 7
36 36 36 36 36 36

1 2 3 4 5 6
2
=
Further, E (X ) 4 ´ + 9 ´ + 16 ´ + 25 ´ + 36 ´ + 49 ´
36 36 36 36 36 36
5 4 3 2 1 1974
64+ ´ + 81´ + 100 ´ + 121´ + 144´ = = 54.8
36 36 36 36 36 36

Thus, Var(X) = 54.8 - 49 = 5.8
(c) The probability distribution of X in example 3 was obtained as

X 1 2 3
4 12 4
p ( )
X
20 20 20
From the above, we can write

4 12 4
=
X
E ( ) 1´ + 2 ´ + 3 ´ = 2
20 20 20
4 12 4
2
=
and ( E X ) 1´ + 4 ´ + 9 ´ = 4.4
20 20 20
\ Var(X) = 4.4 - 4 = 0.4
Expected Monetary Value (EMV)

When a random variable is expressed in monetary units, its expected value is often termed as
expected monetary value and symbolised by EMV.

Example 5: If it rains, an umbrella salesman earns Rs 100 per day. If it is fair, he loses Rs
15 per day. What is his expectation if the probability of rain is 0.3?
Solution.

Here the random variable X takes only two values, X = 100 with probability 0.3 and X = - 15
1 2
with probability 0.7.
Thus, the expectation of the umbrella salesman

= 100 ´ 0.3 - 15 ´ 0.7 = 19.5
The above result implies that his average earning in the long run would be Rs 19.5 per day.

Example 6: A person plays a game of throwing an unbiased die under the condition that
he could get as many rupees as the number of points obtained on the die. Find the expectation
and variance of his winning. How much should he pay to play in order that it is a fair game?
Solution.

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