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nicekikah

Answered question

2021-01-06

If X and Y are random variables and c is any constant, show that

E (c X) = c E (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 (c X) = c E (X)

.
Therefore, for any number c,

E (c X) = c E (X)

.
Conclusion:
Hence, the relation

E (c X) = c E (X)

is proved if X and Y are random variables and c is any constant.

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