Math Is Fun Forum

Mathperson132

Guest

Probability help please

Suppose X is a continuous random variable with density
f(x)=x^3, for 0<x<t and f(x) = 0 otherwise.

Compute the expectation of X.

I know that first you integrate x^3 on the interval from 0 to t, so this gives you

t^4 divided by 4.

But after this, I'm unsure of what to do.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Probability help please

First, you need to multiply your density function by the value it produces.  In other words, you need to integrate x^4, not x^3.

I'm not that great at probability, so I'm using intuition from here on out.  Wikipedia says that the expectation is the integral of x^4 from -infinity to infinity, but this produces an expected value of t^5 / 5, which is greater than t (the maximum value of x) for every t greater than the fourth root of 5.  My guess is to set up 2 integrals equal to each other to find the middle value, like this:

From here we get the equation

Like I said, this is just a guess.  It makes intuitive sense to me, but my intuition is usually wrong when it comes to probability.

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Probability help please

First, you calculate the value of t. You know that the total probability must be 1, therefore

Hence the expected value is

Last edited by JaneFairfax (2007-11-01 12:53:48)