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Probability Question

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose that X has pdf

    f(x) = k[itex]x^{2}[/itex] for -1[itex]\leq x \leq 1[/itex], 0 otherwise

    (a) What is k?
    (b) What is E[itex]x^{n}[/itex] where n[itex]\geq 0[/itex] is an odd integer?
    (c) What is E[itex]x^{n}[/itex] where n[itex]\geq 0[/itex] is an even integer?

    2. Relevant equations



    3. The attempt at a solution

    For (a) I get 1.5
    For (b) I get (-3/8)
    for (c) I get (9/8)

    C is my concern because the expected value for when n is even is greater than one. Is this a problem? My initial thought that it was because x takes on values greater than 1 and have a probability of occurring. I have checked my work and saw nothing wrong. So I just wanted to make sure that my answer to c was impossible before posting my math. It's very simple math and I don't see what I'm doing wrong. Thanks for any help that you can provide.
     
  2. jcsd
  3. Mar 12, 2013 #2

    Ray Vickson

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    Homework Helper

    Show your work. Your answers to (b) and (c) are wrong.
     
  4. Mar 12, 2013 #3
    Using the definition
    $$E[x] = ∫^{\infty}_{-\infty} x f_{x}(x)\, dx$$
    So in the case of raising to a power
    $$E[x^{n}]=∫^{\infty}_{-\infty} x^{n} f_{x}(x)\, dx$$
    We are given that
    $$f_{x}(x) =
    \begin{cases} kx^{2} & \text{for }-1 \leq x \leq 1 \\
    0 & \text{otherwise}
    \end{cases}$$

    In part A, I solved for k and got ##k=\frac{3}{2}##.

    Now for part B
    $$E[x^{n}] = \frac{3}{2} \int^{1}_{-1} x^{n} x^{2}\,dx = \frac{3}{2} \int^{1}_{-1} x^{n+2}\,dx = \frac{3}{2} \left.\frac{x^{n+3}}{n+3}\right|^{1}_{-1} = \frac{3}{2} \left.\frac{x^{n} x^{3}}{n+3}\right|^{1}_{-1} = \frac{3}{2}\left(\frac{1^{n}1^{3}}{n+3}-\frac{(-1)^{n}(-1)^{3}}{n+3}\right) = \frac{3}{2}\left(\frac{1^{n}}{n+3}-\frac{(-1)^{n}(-1)^{3}}{n+3}\right).$$ I think the correct answer is then ##\frac{3}{3+n}## when n even, 0 when n odd.

    I think the first time I did this problem I plugged in 1 and -1 for both n and x and is the reason why I got the answers I did.
     
    Last edited by a moderator: Mar 13, 2013
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