Solved – Regarding the expected value, what’s the meaning of E(-5X) and E(X+Y)

if X∼N(2,1) then the expected value of −5X ? E(−5X) = -5E(X)

if Y~N=(4,1) then the expected value of X+Y? E(X+Y)=E(X)+E(Y)

Can you please tell me what is the meaning of multiplying or summing expected values ?

Assume you roll the dice, if you roll a one you win one euro, if you roll a two then you win 2 euro, … if you roll six than you win six euro.

If the dice is fair then the probability to roll a one is 1/6, the same for rolling a two, … the same for rolling a six.

So for a dice we have $P(X=1)=P(X=2)=dots = P(X=6)=1/6$. If you get one euro if you throw 1, two euro if you get two, … then your ''expected win'' is $1 times 1/6 + 2 times 1/6 + dots + 6 times 1/6=3.5$ which is the expected value of $1X$, i.e. E(1X)=E(X)=3.5$, one because you get 1 euro for each dot that comes up.

If you get 1000 euro for each dot that comes up, then you have to compute the expected values of $1000X$ to compute your expected profit, i.e. $E(1000X)=1000E(X)=3500$.

For sums it is simular, but you use two dice, one with an outcome X, the other one with an outcome Y.

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