Chebyshev's Inequality Problem

[clipperdude21]

Let X be uniformly distributed over (-2,2). Use Chebyshev's inequality to estimate P(abs(x) >= 1) and compare to the exact value.

The answer in the book got something different than what i got so i wanted to see if i was rightEX of a uniformly distrubuted RV is (a+b)/2 so -2+2/0 =0
So i plugged into the formula.

P(abs(x)>=1 <= sigma^2

the variance of a uniformly RV is (b-a)^2/12 so (2-(-2))^2=14 and 14/12 is 4/3.

my first answer ======= P(abs(x)>=1 <= 4/3For the exact part I used the P(X<=x) cdf of a unform RV and got F(1)=3/4. So 1-F(1) = 0.25 which is the probability we are looking for.Did i do this right? Thanks!

 

[EnumaElish]

Write out Cheb's Ineq. and walk through the steps.

 

[tacman]

Yes, you have correctly applied Chebyshev's inequality to estimate the probability P(abs(x) >= 1) and your answer of 4/3 is correct. However, the exact value for this probability is actually 1/2, which is different from the answer in the book. This means that Chebyshev's inequality provides a conservative estimate for this particular problem.