Discrete Random Variable Defined on a Finite Sample Space

1. a) Prove that if X is a discrete random variable defined on a finite sample space, S, then we have E(X) = SUM X(a) P({a}).

b) Use part(a) to prove that if X and Y are discrete random variable defined on the same finite sample space S, then we have E(X + Y) = E(X) + E(Y).

2. Use the result of problem 1 above to show that the average number of heads when a fair coin is flipped n times is equal to n/2. Hint: Let H_i be the number of heads on the ith flip. {Note: compare this to the "brute-force" computation to get an interesting combinatorial identity!]

Solution Preview

Solution:
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1) (a) Let’s denote a random variable as X ~ (X_k), k = 1, n
(p_k)
( ...

Solution Summary

A discrete random variable is expressed in an attached Word document.

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... as 0,1,2,3,4,.... Discrete random variables are usually ... If a random variable can take only a finite number of distinct values, then it must be discrete. ...