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Chapter 8: Q 8.1 (page 390)

Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P{0 < X < 40}?

Short Answer

Expert verified

P[0<X<40]≥1920

Step by step solution

01

Step 1. Given information

Let X is a random variable with mean and variance both 20.

02

Step 2. Denoting mean and variance

E(X) = 20

V(X) = 20

03

Step 3. To find

P[0<X<40]

04

Step 4. Simplification

P[0-20<X-20<40-20]=P[-20<X-20<20]=P[|X-20|<20]

05

Step 5. Using Chebyshev's inequality

1-P[|X-20|≥20]≥1-120=1920

06

Step 6. Final answer 

P[0<X<40]≥1920

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Most popular questions from this chapter

If E[X]=75E[Y]=75Var(X)=10

Var(Y)=12Cov(X,Y)=-3

give an upper bound to

(a) P{|X-Y|>15};

(b) P{X>Y+15};

(c)P{Y>X+15};

A clinic is equally likely to have 2, 3, or 4 doctors volunteer for service on a given day. No matter how many volunteer doctors there are on a given day, the numbers of patients seen by these doctors are independent Poisson random variables with a mean of 30. Let X denote the number of patients seen in the clinic on a given day.

(a) Find E[X]

(b) Find Var(X)

(c) Use a table of the standard normal probability distribution to approximate PP{X>65}.

It Xis a Poisson random variable with a meanλ, showing that fori<λ,

P{X≤i}≤e-λ(eλ)iii

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(a)Give an upper bound for the probability that a student’s test score will exceed85.

(b)Suppose, in addition, that the professor knows that the variance of a student’s test score is equal25. What can be said about the probability that a student will score between 65and85?

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8.5 The amount of time that a certain type of component functions before failing is a random variable with probability density function

f(x)=2x0<x<1

Once the component fails, it is immediately replaced by
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Assuming that the random variables Xi,i≥1,are independent, determine r.
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