PROB_L6

Infinite sequences of Bernoulli trials

The distribution of the sum of $r$ independent, identically distributed geometric random variables with parameter $p$ is negative binomially distributed with parameters $r$ and $p$.

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The probability generating function

Definition 1(Definition).

Let $X$ be a nonnegative integer-valued random variable. The probability generating function ${\Phi }_X$ of $X$ is defined as

$$\begin{array}{r}{\Phi }_X(t)=\sum _{x=0}^{\infty }P(X=x)t^x=\sum _{x=0}^{\infty }{f}_X(x)t^x,-1\le t\le 1\end{array}$$

Probability generating functions of standard distributions

• Let $X$ have a binomial distribution with parameters $n$ and $p$. Then,

$${\Phi }_X(t)=(pt+1-p{)}^n.$$

• Let $X$ have a negative binomial distribution with parameters $\alpha$ and $p$. Then,

$${\Phi }_X(t)={\left(\frac{p}{1-t(1-p)}\right)}^{\alpha }.$$

• Let $X$ have a poisson distribution with parameter $\lambda$. Then,

$${\Phi }_X(t)=e^{\lambda (t-1)}.$$

Theorem 2(Theorem).

Let $X_1,\dots ,X_r$ be independent, nonnegative integer-valued random variables. Then,

$${\Phi }_{X_1+\cdots +X_r}(t)={\Phi }_{X_1}(t)\dots {\Phi }_{X_r}(t).$$

If two non-negative integer valued random variables have the same probability generating function, they must have the same distribution.

Sums of independent random variables

Theorem 3(Theorem).

Let $X_1,\dots ,X_r$ be independent random variables.

1. If $X_i$ has the binomial distribution with parameters $n_i$ and $p$, then $X_1+\cdots +X_r$ has the binomial distribution with parameters $n_1+\cdots +n_r$ and $p$.
2. If $X_i$ has the negative binomial distribution with parameters ${\alpha }_i$ and $p$, then $X_1+\cdots +X_r$ has the binomial distribution with parameters ${\alpha }_1+\cdots +{\alpha }_r$ and $p$.
3. If $X_i$ has the poisson distribution with parameter $\lambda$, then $X_1+\cdots +X_r$ has the poisson distribution with parameter ${\lambda }_1+\cdots +{\lambda }_r$.