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Chapter 4: Q168SE (page 281)

Assume that xhas an exponential distribution withθ=3.

Find

a.P(x≤1)

b.P(x>1)

c.P(x=1)

d.P(x≤6)

e.P(2≤x≤10)

Short Answer

Expert verified
  1. Px≤1=0.9503
  2. role="math" localid="1660277023394" Px>1=0.0497
  3. role="math" localid="1660277013921" Px=1=0.1491
  4. Px≤6=0.999
  5. P2≤x≤10=0.0024

Step by step solution

01

Given information

X is an exponential random variable andθ=3 .

02

Define the probability distribution function

The p.d.f of X is

fx=θe-θx;x≥0=3e-3x

03

Calculate

Px≤1=∫013e-3xdx=3∫013e-3xdx=313-e-33=1-e-3=1-0.0497=0.9503

Hence,Px≤1=0.9503Px≤1=0.9503

04

Calculate P ( x > 1)

Px>1=1-Px≥1=1-3∫01e-3xdx=1-0.9503=0.0497

Hence,Px>1=0.0497

05

Calculate P ( x = 1 ) 

Px=1=3e-3x=3e-3=3×0.0497=0.1491

Hence, Px=1=0.1491Px=1=0.1491

06

Calculate P(x≤6)

Px≤6=3∫06e-3dx=313-e-183=e-18e-18-1=0.999

Hence,Px≤6=0.999

07

Calculate

P2≤x≤10=3∫210e-3xdx=3e-63-e-303=e-30e-24-1=0.0024

Hence,P2≤x≤10=0.0024

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Most popular questions from this chapter

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