A discrete random variable X has a pdf (of the form f(x) = c(8 - x) for
x = 0, 1, 2, 3, 4, 5, and zero otherwise.
(a) Find the constant c.
n f(n)
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0 c(8 - 0) = 8c = 8/33
1 c(8 - 1) = 7c = 7/33
2 c(8 - 2) = 6c = 6/33
3 c(8 - 3) = 5c = 5/33
4 c(8 - 4) = 4c = 4/33
5 c(8 - 5) = 3c = 3/33
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total = 33c = 1
c = 1/33
(b) Find the CDF, F(x).
n F(n)
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0 F(0) = f(0) = 8/33
1 F(1) = F(0)+f(1) = 8/33+7/33 = 15/33
2 F(2) = F(1)+f(2) = 15/33+6/33 = 21/33
3 F(3) = F(2)+f(3) = 21/33+5/33 = 26/33
4 F(4) = F(3)+f(4) = 26/33+4/33 = 30/33
5 F(5) = F(4)+f(5) = 30/33+3/33
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(c) Find P[X > 2]
P[X > 2] = P[X = 3] + P[X = 4] + P[X = 5] = 5/33 + 4/33 + 3/33 = 12/33 = 4/11
or you could do it this way if there were very many fewer this way:
P[X > 2] = 1 - P[x < F(2) = 1 - 21/33 = 33/33 - 21/33 = 12/33 = 4/11
(d) Find E(X).
Edwin