nicekikah

2021-01-06

If X and Y are random variables and c is any constant, show that $E\left(cX\right)=cE\left(X\right)$.

Margot Mill

Approach:
Let X represent any random variable, the probability distribution pattern is as follows,
$E\left(X\right)={x}_{1}{p}_{1}+{x}_{2}{p}_{2}+{x}_{3}{p}_{3}+.....+{x}_{n}{p}_{n}-$
Here, ${x}_{1},{x}_{2},{x}_{3},....{x}_{n}$ are all possible favorable outcomes and ${p}_{1},{p}_{2},{p}_{3},.....{P}_{n}$ are their respective probabilities.
Calculation:
Consider the expected value of random variable X as,
$E\left(X\right)={x}_{1}{p}_{i}+{x}_{2}{p}_{2}+{x}_{3}{p}_{3}+{x}_{n}{p}_{n}$
The associated probabilities will not change of the variable X is changed to eX, where, c is any constant value.
From part (a),
$E\left(c\right)=c$.
Therefore,
$E\left(cX\right)=cE\left(X\right)$.
Therefore, for any number c, $E\left(cX\right)=cE\left(X\right)$.
Conclusion:
Hence, the relation $E\left(cX\right)=cE\left(X\right)$ is proved if X and Y are random variables and c is any constant.

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