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

nicekikah 2021-01-06 Answered
If X and Y are random variables and c is any constant, show that E(cX)=cE(X).
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Expert Answer

Margot Mill
Answered 2021-01-07 Author has 106 answers
Approach:
Let X represent any random variable, the probability distribution pattern is as follows,
E(X)=x1p1+x2p2+x3p3+.....+xnpn
Here, x1,x2,x3,....xn are all possible favorable outcomes and p1,p2,p3,.....Pn are their respective probabilities.
Calculation:
Consider the expected value of random variable X as,
E(X)=x1pi+x2p2+x3p3+xnpn
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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