nicekikah

2021-01-06

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

Margot Mill

Skilled2021-01-07Added 106 answers

Approach:

Let X represent any random variable, the probability distribution pattern is as follows,

$E(X)={x}_{1}{p}_{1}+{x}_{2}{p}_{2}+{x}_{3}{p}_{3}+.....+{x}_{n}{p}_{n}-$

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,

$E(X)={x}_{1}{p}_{i}+{x}_{2}{p}_{2}+{x}_{3}{p}_{3}+{x}_{n}{p}_{n}$

The associated probabilities will not change of the variable X is changed to eX, where, c is any constant value.

From part (a),

$E(c)=c$ .

Therefore,

$E(cX)=cE(X)$ .

Therefore, for any number c,$E(cX)=cE(X)$ .

Conclusion:

Hence, the relation$E(cX)=cE(X)$ is proved if X and Y are random variables and c is any constant.

Let X represent any random variable, the probability distribution pattern is as follows,

Here,

Calculation:

Consider the expected value of random variable X as,

The associated probabilities will not change of the variable X is changed to eX, where, c is any constant value.

From part (a),

Therefore,

Therefore, for any number c,

Conclusion:

Hence, the relation

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