| Line 34: | Line 34: | ||
P(X=x, Y=n-x) | P(X=x, Y=n-x) | ||
=P(X=x)P(Y=n-x)\\ | =P(X=x)P(Y=n-x)\\ | ||
| − | =\frac{e^{-\lambda_1}\lambda^x}{x!}\times \frac{e^{-\lambda_2}\lambda^(n-x)}{(n-x)!} | + | =\frac{e^{-\lambda_1}\lambda^x}{x!}\times \frac{e^{-\lambda_2}\lambda^(n-x)}{(n-x)!} |
=\frac{e^{-(\lambda_1+\lambda_2)}}{x!} | =\frac{e^{-(\lambda_1+\lambda_2)}}{x!} | ||
\left( | \left( | ||
| Line 41: | Line 41: | ||
\end{array} | \end{array} | ||
\right) | \right) | ||
| − | \lambda_1^x\lambda_2^{n-x} | + | \lambda_1^x\lambda_2^{n-x} |
</math> | </math> | ||
| + | |||
| + | Also | ||
| + | |||
<math> | <math> | ||
{P(X+Y=n)} | {P(X+Y=n)} | ||
| Line 56: | Line 59: | ||
&=\frac{e^{-(\lambda_1+\lambda_2)}}{n!}(\lambda_1+\lambda_2)^n | &=\frac{e^{-(\lambda_1+\lambda_2)}}{n!}(\lambda_1+\lambda_2)^n | ||
</math> | </math> | ||
| − | So | + | |
| + | So, we get | ||
<math> | <math> | ||
P(X=x|X+Y=n) = | P(X=x|X+Y=n) = | ||
Revision as of 13:14, 3 December 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2015
First of all, the conditional distribution can be written as:
$ P(X=x|X+Y=n) =\frac{P(X=x, X+Y=n)}{P(X+Y=n)} =\frac{P(X=x, Y=n-x)}{P(X+Y=n)} $
And
$ P(X=x, Y=n-x) =P(X=x)P(Y=n-x)\\ =\frac{e^{-\lambda_1}\lambda^x}{x!}\times \frac{e^{-\lambda_2}\lambda^(n-x)}{(n-x)!} =\frac{e^{-(\lambda_1+\lambda_2)}}{x!} \left( \begin{array}{c} n\\x \end{array} \right) \lambda_1^x\lambda_2^{n-x} $
Also
$ {P(X+Y=n)} ={\sum_{k=0}^{k=n}P(X=k,Y=n-k)}\\ ={\sum_{k=0}^{k=n}P(X=k)P(Y=n-k)}\\ =\frac{e^{-(\lambda_1+\lambda_2)}}{n!}\sum_{k=0}^{k=n} \left( \begin{array}{c} n\\k \end{array} \right) \lambda_1^k\lambda_2^{n-k} &=\frac{e^{-(\lambda_1+\lambda_2)}}{n!}(\lambda_1+\lambda_2)^n $
So, we get $ P(X=x|X+Y=n) = \left( \begin{array}{c} n\\k \end{array} \right) (\frac{\lambda_1}{\lambda_1+\lambda_2})^x(\frac{\lambda_2}{\lambda_1+\lambda_2})^{n-x} \end{align*} $
