# Marginal distribution function

1. Oct 29, 2010

### _joey

Given:

$$P(X=x, Y=y)=\frac{a^ye^{-2a}}{x!(y-x)!}$$ where $$x=0,1,2,...y$$ and $$y=0,1,2...\infty$$, and $$a>0$$

Find $$P(X=x)$$ and $$P(Y=y)$$

An example is provided in a book on books.google.com
Page 96

Here is my attempted solution
$$p_{X}(x)=\Sigma_{y=0}^{\infty}\frac{a^ye^{-2a}}{x!(y-x)!}=e^{-2a}+ae^{-2a}+\frac{a^2e^{-2a}}{x!(2-x)!}+...+\frac{a^ne^{-2a}}{x!(n-x)!}$$

And then I cannot simplify this serie. Any comments and suggestions will be very much appreciated

2. Oct 29, 2010

### _joey

I solved the problem.

3. Oct 29, 2010

### issacnewton

what's the answer you got since your summation over y is wrong... y should go as
x, x+1,x+2...... infty. for given x

4. Oct 29, 2010

### _joey

maybe it is wrong. Elaborate why 'y should go as x, x+1, x+2.....infty'

5. Oct 30, 2010

### _joey

I've got another question: $$E[X|Y=y]=?$$

My answer is $$E[X|Y=y]=\frac{y}{2}e^{2a-2y}$$

I am not sure if it is correct.

Thanks!

Last edited: Oct 30, 2010
6. Oct 30, 2010

### issacnewton

just plot the points (x,y) on the graph.lets say y=0, then x can take value x=0
so the plotted point would be (0,0) . now if y=1 then x=0,1 so the plotted points would be
(0,1) , (1,1). if y=2 then x=0,1,2. so the plotted points would be (0,2),(1,2),(2,2) and so on.
now the marginal distribution for x is for some given x , that means we fix the value of x and then look for the values that y can take. so on this graph, if you fix value of x to be say 5,then y must take values 5 onwards. y can't take value less than 5 because then x=5 would not be possible. if you take value of x to be 7, then y must take values 7 onwards. this is easy to see
if you draw a vertical line from some fixed value of x. you can see that , y can take values
x onwards for that fixed value of x.

once you do that, the summation is very easy.

7. Oct 30, 2010

### issacnewton

now

$$E[X|Y=y]=\sum_{x\in X} x\, \frac{P[X=x,Y=y]}{P[Y=y]}$$

so first you have to calculate $$P(Y=y)$$ here for a fixed value of y, x can take values from 0 to y.

$$P(Y=y)\,=\, \sum_{x=0}^y \, \frac{a^y e^{-2a}}{x!\, (y-x)!}$$

$$= a^y e^{-2a}\, \sum_{x=0}^y \, \frac{1}{x!(y-x)!}$$

now here you do a small trick. you multiply and divide by y!. since y! is not dependent on x, you pull out the denominator y! and you have y! in the numerator. then what you have is
a binomial coefficient.

$$\sum_{x=0}^y \, \binom{y}{x}=\, \sum_{x=0}^y \, \frac{y!}{x!(y-x)!}$$

the above sum is just $$2^y$$. so

$$P(Y=y)\, = \, \frac{a^y e^{-2a}}{y!} \, 2^y$$

then use this to do the required summation. with the above expression, final answer I get is

$$E[X|Y=y]=\frac{y}{2}$$