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

### Relevant Questions If X and Y are random variables and c is any constant, show that $$E(X+Y)=E(X)+E(Y)$$. If X and Y are random variables and c is any constant, show that $$E(X-Y)=E(X)-E(Y)$$. If X and Y are random variables and c is any constant, show that $$E(c)=c$$. Let X and Y be Bernoulli random variables. Let $$Z = XY$$.
a) Show that Z is a Bernoulli random variable.
b) Show that if X and Y are independent, then $$P_{Z} = P_{X}P_{Y}$$. if $$(X_{1},...,X_{n})$$ are independent random variables, each with the density function f, show that the joint density of $$(X_{(1)},...,X_{(n)}$$ is n!f $$(x_{1})..f(x_{n}), x_{1} asked 2020-12-30 Consider a set of 20 independent and identically distributed uniform random variables in the interval (0,1). What is the expected value and variance of the sum, S, of these random variables? Select one: a.\(\displaystyle{E}{\left({S}\right)}={10}$$, $$\displaystyle{V}{a}{r}{\left({S}\right)}={1.66667}$$
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c.$$\displaystyle{E}{\left({S}\right)}={100}$$, $$\displaystyle{V}{a}{r}{\left({S}\right)}={200}$$
a.$$\displaystyle{E}{\left({S}\right)}={10}$$, $$\displaystyle{V}{a}{r}{\left({S}\right)}={12}$$ The expectation of the number of people getting their correct hats in the hat check problem, using linearity of expectations.
n persons receive n hats in a random fashion.
Formula used: By linearity of expectation function,
If $$X_{1},X_{2},...,X_{n}$$ are random variables, then
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If $$\displaystyle{X}_{{{1}}}{\quad\text{and}\quad}{X}_{{{2}}}$$ both have a variance of 2.0, what is the variance of L? Suppose that $$X_{1}, X_{2}, ..., X_{200}$$ is a set of independent and identically distributed Gamma random variables with parameters $$\alpha = 4, \lambda = 3$$. 