# poisson and negative binomial

• Nov 19th 2008, 08:01 AM
chrisc
poisson and negative binomial
If X has Poisson distribution with the mean of 2.3, Y has Negative Binomial
distribution with p = 0.62 and k = 4, and X and Y are independent,
find Pr(X + Y > 5).
Hint: Use probability generating function.

I would usually fill in on what I have tried so far, but even with the hint here I can't seem to get started. Any help is appreciated
• Nov 19th 2008, 02:31 PM
mr fantastic
Quote:

Originally Posted by chrisc
If X has Poisson distribution with the mean of 2.3, Y has Negative Binomial
distribution with p = 0.62 and k = 4, and X and Y are independent,
find Pr(X + Y > 5).
Hint: Use probability generating function.

I would usually fill in on what I have tried so far, but even with the hint here I can't seem to get started. Any help is appreciated

Let $\displaystyle U = X + Y$. Note that $\displaystyle U = 4, 5, 6, \, ....$

$\displaystyle \Pr(U > 5) = 1 - \Pr(U \leq 5) = 1 - \Pr(U = 4) - \Pr(U = 5)$.

Let $\displaystyle g_U(s)$ be the pgf of U.

Then $\displaystyle g_U(s) = g_X(s) \cdot g_Y(s)$ (since X and Y are independent) where $\displaystyle g_X(s)$ is the pgf of X and $\displaystyle g_Y(s)$ is the pgf of Y.

And you should know that $\displaystyle g_X^{(n)}(0) = n! \Pr(U = n)$.
• Nov 19th 2008, 11:41 PM
mr fantastic
Quote:

Originally Posted by mr fantastic
Let $\displaystyle U = X + Y$. Note that $\displaystyle U = 4, 5, 6, \, ....$

$\displaystyle \Pr(U > 5) = 1 - \Pr(U \leq 5) = 1 - \Pr(U = 4) - \Pr(U = 5)$.

Let $\displaystyle g_U(s)$ be the pgf of U.

Then $\displaystyle g_U(s) = g_X(s) \cdot g_Y(s)$ (since X and Y are independent) where $\displaystyle g_X(s)$ is the pgf of X and $\displaystyle g_Y(s)$ is the pgf of Y.

And you should know that $\displaystyle g_X^{(n)}(0) = n! \Pr(U = n)$.

Perhaps an easier way (but not using the pgf) would be to note that

$\displaystyle \Pr(U = 4) = \Pr(X = 0) \cdot \Pr(Y = 4)$

$\displaystyle \Pr(U = 5) = \Pr(X = 0) \cdot \Pr(Y = 5) + \Pr(X = 1) \cdot \Pr(Y = 4)$

since X and Y are independent.