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Given that
E(aX+b)=aE(X)+b, where
E(X) is the expected value of a discrete random variable
X and
a and
b are constants.
Prove that
Var(aX+b)=a^(2)Var(X)

Given that $E(a X+b)=a E(X)+b$ , where $E(X)$ is the expected value of a discrete random variable $X$ and $a$ and $b$ are constants. $\newline$ Prove that $\operatorname{Var}(a X+b)=a^{2} \operatorname{Var}(X)$

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Find derivatives of sine and cosine functions

Full solution

Q. Given that

E(a X+b)=a E(X)+b

, where

E(X)

is the expected value of a discrete random variable

X

and

a

and

b

are constants.

\newline

Prove that

\operatorname{Var}(a X+b)=a^{2} \operatorname{Var}(X)

Define variance formula: Step $1$ : Define the variance formula for a random variable $X$ . $\text{Var}(X) = \text{E}(X^2) - (\text{E}(X))^2$
Apply expected value formula: Step $2$ : Apply the expected value formula to $aX + b$ . $\newline$ $E(aX + b) = aE(X) + b$
Calculate $E((aX + b)^2)$ : Step $3$ : Calculate $E((aX + b)^2)$ . $\newline$ $E((aX + b)^2) = E(a^2X^2 + 2abX + b^2)$
Expand expected value: Step $4$ : Expand the expected value using linearity. $\newline$ $E(a^2X^2 + 2abX + b^2) = a^2E(X^2) + 2abE(X) + E(b^2)$
Calculate $E(b^2)$ : Step $5$ : Since $b$ is a constant, $E(b^2) = b^2$ . $\newline$ $E(b^2) = b^2$
Substitute to find $E((aX + b)^2)$ : Step $6$ : Substitute back to find $E((aX + b)^2)$ . $\newline$ $E((aX + b)^2) = a^2E(X^2) + 2abE(X) + b^2$
Calculate $\text{Var}(aX + b)$ : Step $7$ : Calculate $\text{Var}(aX + b)$ using the variance formula. $\newline$ $\text{Var}(aX + b) = \text{E}((aX + b)^2) - (\text{E}(aX + b))^2$
Substitute values: Step $8$ : Substitute the values from Steps $2$ and $6$ . $\newline$ $\text{Var}(aX + b) = (a^2\text{E}(X^2) + 2ab\text{E}(X) + b^2) - (a\text{E}(X) + b)^2$
Expand $(aE(X) + b)^2$ : Step $9$ : Expand $(aE(X) + b)^2$ . $(aE(X) + b)^2 = a^2(E(X))^2 + 2abE(X) + b^2$
Simplify $\text{Var}(aX + b)$ : Step $10$ : Simplify $\text{Var}(aX + b)$ . $\newline$ $\text{Var}(aX + b) = a^2\text{E}(X^2) + 2ab\text{E}(X) + b^2 - a^2(\text{E}(X))^2 - 2ab\text{E}(X) - b^2$
Cancel out terms: Step $11$ : Cancel out terms. $\newline$ $\text{Var}(aX + b) = a^2\text{E}(X^2) - a^2(\text{E}(X))^2$
Recognize Var(X) expression: Step $12$ : Recognize the expression for Var(X). $\newline$ Var( $aX + b$ ) = $a^2$ (Var(X))

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