Expectation

**thomas_donald** · December 1st 2009, 03:25 PM

I am wondering if I could get help on the following HW problem:
3 trails each have the same probability of success. If X is the total # of successes and if E[X]=1.8. What is the smallest possible value of P(X=3)?
I know that the answer is 0 but I do not know why. I'll appreciate any help.

**ANDS!** · December 1st 2009, 04:07 PM

Can you post the exact question?

**CaptainBlack** · December 1st 2009, 11:32 PM

Originally Posted by thomas_donald

I am wondering if I could get help on the following HW problem:
3 trails each have the same probability of success. If X is the total # of successes and if E[X]=1.8. What is the smallest possible value of P(X=3)?
I know that the answer is 0 but I do not know why. I'll appreciate any help.

Let $p$ be the probability of success on a single trial. Then:

$E(X)=0.P(x=0)+1.P(x=1)+2.P(x=2)+3.P(x=3)=1.8\ \ \ \ ...(1)$

where:

$P(x=0)=(1-p)^3$ ,

$ P(x=1)=3.(1-p)^2.p $ ,

$ P(x=2)=3.(1-p).p^2 $ ,

$ P(x=3)=p^3 $ .

Substituting these into the equation $(1)$ gives a cubic in $p$ and the question wants to know the smallest positive root of this cubic.

CB

**thomas_donald** · December 2nd 2009, 08:18 AM

Thanks a lot for the help.
This is the exact question and the trails are not independent.
I found the answer.
Thanks again.

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