Finding Formulas for c, μx, and Var(X) in a Center of Mass Word Problem

[PCSL]

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

I know Var(X)=∫(x-μ_x)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.

 

[Mark44]

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

I know Var(X)=∫(x-μ_x)²f(x)dx

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

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.

 

[LCKurtz]

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

I know Var(X)=∫(x-μ_x)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.

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

 

[PCSL]

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

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.

 

[Dick]

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

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.

 

[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 μ_x when it isn't even in the formula? Thanks, and sorry I'm not providing more work but I'm pretty lost...

 

[Dick]

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 μ_x when it isn't even in the formula? Thanks, and sorry I'm not providing more work but I'm pretty lost...

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.

 

[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...?

 

[Dick]

I got (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...?

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.

 

[PCSL]

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.

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.

 

[Mark44]

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