Concept:
If X be a random variable with a finite number of outcomes x1, x2, x3 .... occurring with probabilities p1, p2, p3....... respectively, the expectation of X is defined as:
\(E\left[ x \right] = \mathop \sum \limits_{i = 1}^k {x_i} \cdot {p_i} = {x_1}{p_1} + {x_2}{p_2} \ldots \)
Application:
It is given that the two sides of a fair coin are labelled as 0 and 1 and the coin is tossed two times independently.
∴ The total possible outcomes can be:
O = {(1,1), (1,0), (0,1), (0,0)}
The random variable X is defined as:
X = min (M, N)
So,
X1 = min {(1, 1)} = 1
X2 = min {(1, 0)} = 0
X3 = min {(0, 1)} = 0
X4 = min {(0, 0)} = 0
The probability of occurrence of ‘1’ will be:
\(P\left( 1 \right) = \frac{1}{4}\)
And the probability of occurrence of ‘0’ will be:
\(P\left( 0 \right) = \frac{3}{4}\)
We know that:
\(E\left[ x \right] = \mathop \sum \limits_{i = 1}^k {x_i} \cdot {P_i}\)
\(E\left[ x \right] = 0 \times \frac{3}{4} + 1 \times \frac{1}{4}\)
\(E\left[ x \right] = 0.25\)