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From: grimmjow on 6 Nov 2007 03:27
How would one go about completing the following 2 problems:

1. Suppose that a random variable has a beta distribution with A = 0,
B > 0, α = 5,
and β = 4. Find E(X^3 ) and E(1/X).

2. Let X1, X2 and X3 represent the times necessary to perform three
successive repair
tasks at a certain service facility. Suppose they are independent
normal random variables
with expected values μ1, μ2, and μ3 and variances σ1, σ2,and σ3
respectively. If μ1 = 17, μ2 = 23, μ3 = 29, σ1 = 9, σ2 = 16, and σ3 =
25, calculate P(X1 + 2X2 ≥ 3X3), and P(X1 + 2X2 ≥ 63/29X3).

^ = to the power of

any help would be helpful!

From: Gus Gassmann on 6 Nov 2007 06:56
On Nov 6, 8:27 am, grimmjow <grimm.or....(a)gmail.com> wrote:
> How would one go about completing the following 2 problems:
>
> 1. Suppose that a random variable has a beta distribution with A = 0,
> B > 0, α = 5,
> and β = 4. Find E(X^3 ) and E(1/X).

You know the probability density of the beta distribution (or you can
look it up), and you should be able to write down the integral for
each expected value. After that it is a matter of grinding out the
integral, which is going to be a polynomial at worst (if you can't use
other simplifying tricks).

> 2. Let X1, X2 and X3 represent the times necessary to perform three
> successive repair
> tasks at a certain service facility. Suppose they are independent
> normal random variables
> with expected values μ1, μ2, and μ3 and variances σ1, σ2,and σ3
> respectively. If μ1 = 17, μ2 = 23, μ3 = 29, σ1 = 9, σ2 = 16, and σ3 =
> 25, calculate  P(X1 + 2X2 ≥ 3X3), and P(X1 + 2X2 ≥ 63/29X3).

The sum of normal random variables is normal, as is the difference, so
rewrite the first probability as
P(X1 + 2X2 - 3X3 >= 0). You'll have to find the mean and variance of
this expression in terms of
μ1, μ2, μ3, σ1, σ2,and σ3, and then it should be straightforward.

> ^ = to the power of
>
> any help would be helpful!


From: Ray Vickson on 6 Nov 2007 11:25
On Nov 6, 12:27 am, grimmjow <grimm.or....(a)gmail.com> wrote:
> How would one go about completing the following 2 problems:
>
> 1. Suppose that a random variable has a beta distribution with A = 0,
> B > 0, α = 5,
> and β = 4. Find E(X^3 ) and E(1/X).

Write down the integrals and perform them. Since alpha and beta are
positive integers you will get a simple polynomial to integrate in
each case. It will be a bit messy and it will take some time and
effort, but will be perfectly straightforward.

>
> 2. Let X1, X2 and X3 represent the times necessary to perform three
> successive repair
> tasks at a certain service facility. Suppose they are independent
> normal random variables
> with expected values μ1, μ2, and μ3 and variances σ1, σ2,and σ3

Are these the standard deviations? You say they are the variances, but
your notation violates the usual practice that writes variance as
sigma^2, not as sigma.


> respectively. If μ1 = 17, μ2 = 23, μ3 = 29, σ1 = 9, σ2 = 16, and σ3 =
> 25, calculate P(X1 + 2X2 ≥ 3X3), and P(X1 + 2X2 ≥ 63/29X3).

Given independent normal random variables, what are the distributions
of Z1 = X1 + 2X2 - 3X3 and of Z2 = X1 + 2X2 - (63/29)X3? How can you
find their means and variances? These questions are answered in any
decent textbook, and possibly in your course notes.

>
> ^ = to the power of
>
> any help would be helpful!

Best advice: sit down with a pencil and some paper, open your notes
and/or textbook, and get to work.

R.G. Vickson
Adjunct Professor, University of Waterloo


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