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

Question
Random variables
asked 2021-01-06
If X and Y are random variables and c is any constant, show that \(E(cX)=cE(X)\).

Answers (1)

2021-01-07
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.
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