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# Discrete random variables

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1. I don't understand this symmetry thing... how did they find E(x) without working out the unknowns??
2. Remember that the probablity will equal to one.Therefore, a+b+a=1. From here you will get simultaneous equations, find the value of a and b, then work your way to find E(X).
3. (Original post by FamilyFirst)
I don't understand this symmetry thing... how did they find E(x) without working out the unknowns??
undercxver has correctly pointed out that you can solve this question by solving the simultaneous equations that fall out from the information you are given.

However, I take it that you're puzzled by the gnomic "by symmetry" in the answer for E[X]. The point here is that the probability distribution is symmetric about the value X=2. Whenever a distribution is symmetric about a value you can immediately write down the value of the mean (*). Why?

To make this easier to see, think about the case where X takes the values -1, 0 and 1 and the distribution is still a, b, a. Then E[X] = (-1).a +(0).b + (1).a = 0. Can you see how the cancellation occurs? Now extend this to the case where X takes on more values: -3,-2,-1,0,1,2,3, for example, but where the distribution is still symmetric. Do you see that the part of the expectation sum where X is negative cancels with the part where it is positive?

Now see if you can generalize this to where the distribution is symmetrical about a non-zero value of X.

(*) for the experts: provided that the mean exists.

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