Question

Find the mean variance and standard deviation of the following probability distribution 

xi :	a	b
pi :	p	q

where p + q = 1 .

Answer 1

xi	pi	pixi	pixi2
a	p	ap	a2p
b	q	bq	b2q
		`∑`pixi = ap + bq	`∑`pixi2=a2p+b2q

\[\text{ Now } , \]
\[\text{ Mean }  = \sum p_i x_i = ap + bq\]
\[\text{ Variance}  = \sum p_i {x_i}2^{}_{} - \left( \text{Mean } \right)^2 = a^2 p + b^2 q - \left( ap + bq \right)^2 \]
\[ = a^2 p + b^2 q - a^2 p^2 - b^2 q^2 - 2abpq\]
\[ = a^2 p - a^2 p^2 + b^2 q - b^2 q^2 - 2abpq\]
\[ = a^2 p\left( 1 - p \right) + b^2 q\left( 1 - q \right) - 2abpq\]
\[ = a^2 pq + b^2 qp - 2abpq ............... \left( \because p + q = 1 \right)\]
\[ = pq\left( a^2 + b^2 - 2ab \right)\]
\[ = pq \left( a - b \right)^2 \]
\[\text{ Step Deviation } = \sqrt{\text{ Variance} }\]
\[ = \sqrt{pq \left( a - b \right)^2}\]
\[ = \left| a - b \right|\sqrt{pq}\]