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  • Provide Solution For R.D. Sharma Maths Class 12 Chapter 31 Mean and Variance of a random Exercise 31.2 Question 3 Maths Textbook Solution.

Provide Solution For  R.D. Sharma Maths Class 12 Chapter 31 Mean and Variance of a random Exercise 31.2 Question 3 Maths Textbook Solution.

Answers (1)

Answer:

ap+bq,\left ( a^{2}-b^{2} \right )pq,\mid a-b\mid pq

Hint:

E\left ( X \right )=ap+bq

V\left ( X \right )=\left ( a-b \right )^{2}pq

Standard  deviation=\mid a-b\mid \sqrt{pq}

Given:

\begin{aligned} &\begin{array}{|c|c|c|} \hline x_{i} & a & b \\ \hline p_{i} & p & q \\ \hline \end{array}\\ &\text { Where } p+q=1 \end{aligned}

Solution:

\begin{aligned} &\text { Mean } E(X)=\sum_{i=1}^{n} x_{i} p_{i} \\ \end{aligned}

\begin{aligned} &=a \times p+b \times q \\ \end{aligned}

\begin{aligned} &=a p+b q \\ &E\left(X^{2}\right)=\sum_{i=1}^{n} x_{i}^{2} p_{i} \\ \end{aligned}

\begin{aligned} &=a^{2} \times p+b^{2} \times q \\ &=a^{2} p+b^{2} q \\ \end{aligned}

\begin{aligned} &V(X)=E\left(X^{2}\right)-(E(X))^{2} \\ &=\left(a^{2} p+b^{2} q\right)-(a p+b q)^{2} \\ &=a^{2} p+b^{2} q-a^{2} p^{2}-b^{2} q^{2}-2 a b p q \\ &=a^{2} p(1-p)+b^{2} q(1-q)-2 a b p q \\ &=a^{2} p q+b^{2} p q-2 a b p q \\ &=(a-b)^{2} p q \\ &\text { Standard deviation }=\sqrt{V(X)}=\sqrt{(a-b)^{2} p q}=|a-b| \sqrt{p q} \end{aligned}

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