Page 255 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 6. Mathematical definition of probability, was given by J. Bernoulli.
7. Classical definition is also known as ‘a priori’ definition of probability.
8. Two or more outcomes of an experiment are said to be mutually exclusive if the occurrence
of one of them precludes the occurrence of all others in the same trial i.e. they cannot occur
jointly.

12.2 Theorems on Expectation

Theorem 1:
Expected value of a constant is the constant itself, i.e., E(b) = b, where b is a constant.
Theorem 2:
E(aX) = aE(X), where X is a random variable and a is constant.

12.2.1 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: If it rains, an umbrella salesman earns 100 per day. If it is fair, he loses 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, X1 = 100 with probability 0.3 and X2 = – 15
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 19.5 per day.

12.2.2 Expectation of the Sum or Product of two Random Variables

Theorem 1:
If X and Y are two random variables, then E(X + Y) = E(X) + E(Y).
Theorem 2:

If X and Y are two independent random variables, then
E(X.Y) = E(X).E(Y)

12.2.3 Expectation of a Function of Random Variables

Let X ,Y be a function of two random variables X and Y. Then we can write

m n
E X ,Y X ,Y p
i j ij
i 1 j 1

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