STAT230 Study Guide - Midterm Guide: Negative Binomial Distribution, Cumulative Distribution Function, Rush Hour

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14 Nov 2012
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1. Let Xbe a random variable that takes possible values at non-negative integers and has cumulative
distribution function
F(x) = P(Xx)=12xfor x= 0,1,2,· · ·
(a) Find the probability function of X.
Soln:f(x) = P(X=x) = F(x)F(x1) = 2(x1) 2x= 2x,x= 1,2,· · ·.
f(0) = 0, because F(0) = 0.
(b) Find P(X= 5) and P(X5)
Soln:P(X= 5) = f(5) = 25;
P(X5) = 1 P(X4) = 1 F(4) = 1 (1 24) = 24.
Or: P(X5) = P
x=5 2x= 24.
(c) Find P(X= 5 |X5)
Soln:
P(X= 5 |X5) = P(X= 5, X 5)
P(X5) =P(X= 5)
P(X5) =25
24= 0.5
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