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Letc>0and0≤α≤1.Also let X, Y, and Tbe random variables.

(a) If P(X>c)=α, determine P(X≤c) in terms of α.

(b) Suppose that P(Y>c)=α2andP(Y<-c)=P(Y>-c). Determine P(-c≤Y≤c) in terms of α.

(c) Suppose that P(-c≤T≤c)=1-αand also suppose that P(T<-c)=PT>c. FindPT>c in terms of α.

Short Answer

Expert verified

Part (a)P(X≤c)=1-α.

Part (b) P(-c≤Y≤c)=1-α.

Part (c) localid="1651633688867" P(T>c)=α2.

Step by step solution

01

Part (a) Step 1. Given information

P(X>c)=α

02

Part (a) Step 2. Solution

P(X≤c)=1-P(X>c)=1-α

03

Part (b) Step 1. Given information 

P(Y>c)=α2andP(Y<-c)=P(Y>-c)

04

Part (b) Step 2. Solution

P(-c≤Y≤c)=1-[P(Y<-c)+P(Y>-c)]=1-[2P(Y<-c)]=1-2α2=1-α

05

Part (c) Step 1. Given information

P(-c≤Y≤c)=1-αandP(T>c)=P(T<-c)

06

Part (c) Step 2. Solution

P(T>c)=1-[P(-c≤Y≤c)+P(T<-c)]P(T>c)=1-[1-α+P(T>c)]2P(T>c)=αP(T>c)=α2

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