# Center of Mass Word Problem

1. Oct 23, 2011

### PCSL

Let p be a positive constant. Suppose that a random variable X has probability function f(x)=cxp(1-x) for 0≤x≤1. Find formulas for c, μx, and Var(X) in terms of p.

I know Var(X)=∫(x-μx)2f(x)dx

I know that I did not show any work so please just give me a hint (I'm not asking you to solve it for me). Thanks and I would have provided work if I even had a guess on where to start.

P.S. I'm kind of confused why this is at the end of the center of mass chapter since I totally understood everything up to this problem.

2. Oct 23, 2011

### Staff: Mentor

Since f(x) is a probability function, isn't its integral over the interval equal to 1?
Also, isn't μX the same as E(X)? And isn't this an integral as well?

3. Oct 23, 2011

### LCKurtz

Why have you titled this as a "Center of Mass Word Problem"?

4. Oct 23, 2011

### PCSL

Because it is a word problem at the end of the section titled center of mass. The solution involves finding the second moment also, I believe.

@Mark I did not know that Mu sub x is equivalent to E(X). I'll go on that assumption and see if I get the right answer.

5. Oct 23, 2011

### Dick

Oh, finding the expectation value of f(x) could be considered as a 'center of mass' problem. The formula looks similar. It's a first moment. But the title is not that important. The first thing PCSL should do is find c. Then the rest should be easy.

6. Oct 23, 2011

### PCSL

Alright, so as Mark said

$$\int_{0}^{1} cx^p(1-x)dx=1$$
because it is a PDF

When solving this for c do I assume that c is a constant and pull it out of the integral? How do I solve this for μx when it isn't even in the formula? Thanks, and sorry I'm not providing more work but I'm pretty lost...

7. Oct 23, 2011

### Dick

Sure, c is a constant. Pull it out and solve for it in term of p. Then use that value of c to find mu_x. Etc.

Last edited: Oct 23, 2011
8. Oct 23, 2011

### PCSL

I got $$[\frac{cx^{p+1}}{p+1}-\frac{cx^{p+2}}{p+2}]_{0}^{1}$$
which I simplified to
(p+1)(p+2)=c

What do you mean I use this value of c to find mu sub x. Actually, it says above mu_x is equal to x so is that relevant...?

9. Oct 23, 2011

### Dick

Ok, so f(x)=(p+1)(p+2)x^p(1-x). Now you just integrate f(x)*x to find E(x). Then use that find Var(f(x)). It should be routine from here on.

10. Oct 23, 2011

### PCSL

Thank you so much! I assume I continue to integrate from 0 to 1 and I can pull (p+1)(p+2) out of the integral? Thanks again.

11. Oct 23, 2011

### Staff: Mentor

Yes, the limits of integration for all integrals are 0 and 1. And for the integration, p is a constant.