Question

Anyone available for Stat Homework that's due tomorrow? Gamma distribution - pasting the image of the problem in the comments. Thank you in advance!!

Answer 1

$$
\Large E(X) = \int x \cdot f(x)

$$

Answer 2

is that the same though, as \[E(\frac{ 1 }{ X })\]?

Answer 3

and what is x? f(x) is my gamma function, correct? Is x=6?

Answer 4

almost the same

Answer 5

$$
\Large E(\frac1X) = \int\frac1 x \cdot f(x)

$$

Answer 6

$$
\Large E(\frac1X) = \int_{-\infty }^{\infty} ~ \frac1 x \cdot \frac{1}{\Gamma (6)\cdot 2^6 }\cdot x^{6-1}e^{\frac{-x}{2}}

$$

Answer 7

HI
not to butt in, but i think there might be a way to do this directly
also, isn't the integral from 0 to \(\infty\)? gamma not defined for \(x<0\) if i am not mistaken

Answer 8

check the formula for the kth moment on page 36 here
http://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2006/lecture-notes/lecture6.pdf

Answer 9

looks like it might be t
\[\frac{\Gamma(5)}{\Gamma(6)\frac{\pi}{2}^{-1}}\]

Answer 10

\[\Gamma(5)=4!, \Gamma(6)=5! \] so it might be just
\[\frac{\pi}{5}\]

Answer 11

nope bad algebra, \[\frac{\pi}{10}\]

Answer 12

@perl look reasonable or am i way off base?

Answer 13

oh another mistake, i completely ignored the \(2^6\) in the denominator

Answer 14

right, start at x = 0

Answer 15

$$
\Large E(\frac1X) = \int_{0 }^{\infty} ~ \frac1 x \cdot \frac{1}{\Gamma (6)\cdot 2^6 }\cdot x^{6-1}e^{\frac{-x}{2}}

$$

Answer 16

lol i put a \(\pi\) there instead of a 2

Answer 17

it is one

Answer 18

\[\gamma(6)=5!, 5!\times 2^6=7680\]

Answer 19

\Gamma

Answer 20

yeah that one

Answer 21

so wolfram says E(1/x) = .10
http://www.wolframalpha.com/input/?i=int%281%2Fx*x^5*exp%28-%281%2F2%29*x%29%2F%28gamma%286%29*2^6%29%2C+x+%3D+0+..+infinity%29

Answer 22

\[\frac{1}{7680}\int_0^{\infty}x^4e^-{\frac{x}{2}}\]

Answer 23

damn i can't see what i am typing, lotsa typos

Answer 24

\[\frac{1}{7680}\int_0^{\infty}x^4e^{\frac{x}{2}}dx\]

Answer 25

aarrgghh

Answer 26

what about the (1/X) being multiplied in that integral?

Answer 27

\[\frac{1}{7680}\int_0^{\infty}x^4e^{-\frac{x}{2}}dx\]

Answer 28

that turns \(x^5\) in to \(x^4\)

Answer 29

oh, ok I see that

Answer 30

you can do integration by parts a few times

Answer 31

lol yeah quite a few

Answer 32

bet we can do it quicker

Answer 33

\[\frac{\Gamma(5)}{\Gamma(6)2^{-1}}\]

Answer 34

no that is not working dang
i wonder why...

Answer 35

ooh because i am an idiot!!

Answer 36

\[\frac{\Gamma(5)}{\Gamma(6)(\frac{1}{2})^{-1}}\]

Answer 37

\[\frac{4!}{2\times 5!}\]

Answer 38

yayyy!!!!
wasn't that easy?

Answer 39

haha! Thank you to both of you for jumping in to help me!! I am totally with you until the end @misty1212 - I have no idea where that last equation came from. :)

Answer 40

ohhh...we haven't covered moments yet. Guess that's my problem. :(

Answer 41

damn it
\[E(X^k)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)\beta^k}\]

Answer 42

in your case \(\alpha=6, \beta=\frac{1}{2},k=-1\)

Answer 43

and
[\Gamma(\alpha)=(\alpha-1)!\] for integer \(\alpha\)

Answer 44

is that formula given in the pdf

Answer 45

\[\Gamma(\alpha)=(\alpha-1)!\]

Answer 46

damn i hate not having the previewq

Answer 47

sure is a lot easier than integrating by parts 4 times !!!

Answer 48

yeah that is the formula in the pdf, take a look it is derived there

Answer 49

with that formula takes like two seconds if you do it correctly and not mess about like i did 4 or 5 times

Answer 50

You guys are awesome! Thank you so much!

Answer 51

yes thats pretty nifty :)

Answer 52

$$
\Large {
\mathbb{E}(X^k) = \frac{\Gamma(\alpha+k)}{\Gamma(\alpha)\beta^k}

}

$$

Answer 53

oh by the way, lowercase \gamma is used for euler's constant. thats why i was getting a strange answer on maple

Answer 54

I think I need to show my homework with the integration by parts, however I've done this at least 4 times and never get the 1/10 that Wolfram shows. I keep getting -4

Answer 55

you can expedite the integration by parts by making a table

Answer 56

@misty1212 I'm surprised this formula is not mentioned in the wikipedia article.
Unless i missed it
http://en.wikipedia.org/wiki/Gamma_distribution

Answer 57

@BrighterDays want to make a table?

Answer 58

I don't know about making tables?? :)