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

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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(X≤x)=1−2−xfor x= 0,1,2,· · ·

(a) Find the probability function of X.

Soln:f(x) = P(X=x) = F(x)−F(x−1) = 2−(x−1) −2−x= 2−x,x= 1,2,· · ·.

f(0) = 0, because F(0) = 0.

(b) Find P(X= 5) and P(X≥5)

Soln:P(X= 5) = f(5) = 2−5;

P(X≥5) = 1 −P(X≤4) = 1 −F(4) = 1 −(1 −2−4) = 2−4.

Or: P(X≥5) = P∞

x=5 2−x= 2−4.

(c) Find P(X= 5 |X≥5)

Soln:

P(X= 5 |X≥5) = P(X= 5, X ≥5)

P(X≥5) =P(X= 5)

P(X≥5) =2−5

2−4= 0.5