Q.In a business venture, a man can make a profit of $\text{₹ } 2{,}000$ with a probability of $0.4$ or have a loss of $\text{₹ } 1{,}000$ with a probability of $0.6$. What is his expected, variance and standard deviation of profit ?

Expectation is linear, so $E(6x)=6E(x)$.

A key property of mathematical expectation is that a constant factor pulls out of the expectation. For any constant $a$,

$E(aX) = a\,E(X).$

With $a=6$:

$E(6X) = 6\,E(X).$

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$E(X-Y)=E(X)-E(Y)=5-(-2)=7$.

By the linearity of mathematical expectation,

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Expectation is linear and $E(X)$ is a constant, so the mean deviation $E[X-E(X)]=0$.

Let $\mu=E(X)$, which is a constant. Using linearity of expectation:

$E[X-E(X)]=E(X)-E\big(E(X)\big)=E(X)-E(X)=\mu-\mu=0.$ …

Mathematical expectation is $\sum x\,p(x)$; here $10\times\tfrac{1}{10}=1$.

The expected value of a discrete random variable is

$E(X)=\sum x\,p(x).$

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