Discrete Variable

Binomial distribution

Definition

The typical example is binomial distribution. Before we deal with the binomial distribution, we first have to define the bernoulli distribution.

$$X(success)=1 , \quad X(failure)=0 \\ p(x)=p^x(1-p)^{1-x}, x=0,1$$

A sequence of bernoulli trial makes binomial distribution.

$E(X)=p, \; Var(X)=p(1-p), \; M(t)=pe^t+q$

When X is the number of successes in n independent Bernoulli trials, we say that X follows Binomial distribution.

Properties

$$f(X)=\binom n x p^x(1-p)^{n-x} \\ E(X)=np , \; Var(X)=np(1-p) \\ M(t)=(pe^t+q)^n, \; -\infty<t<\infty \\ X \equiv Z_1 +\cdots +Z_n , \; Z_i \sim iid \; Bernoulli(p)$$

Example

Let the winning rate of a gambler be 2 over 3. X is the number of winning for 3 games.

$$p(x)=_3C_x(\dfrac{2}{3})^x(\dfrac{1}{3})^{3-x}, x=0,1,2,3 \\ X \sim B(3,\frac{2}{3})$$

Negative Binomial distribution

Definition

• Number of success : $r$
• Number of failures: $y$
• The rate of success: $p$

$Y$ is equal to the number of failures until $r_{th}$successes.

$$p(y)=_{y+r-1}C_{r-1} p^{r-1}(1-p)^y*p \\ Y \sim NB(r,p)$$

Poisson distribution

$$X \sim Poisson(\lambda), \quad \lambda >0 \\ f(x;\lambda)=\dfrac{\lambda^xe^{-\lambda}}{x!}, \quad x=0,1,2,\cdots \\ P(k \; events \; in \; time \; period)=e^{-\frac{events}{time}*time\;period}\dfrac{(\frac{events}{time}*time \; period)^k}{k!}$$

$\lambda$is the occurrence number of events per time unit.

https://towardsdatascience.com/the-poson-distribution-and-poisson-process-explained-4e2cb17d459

Properties

$$E(X)=Var(X)=\lambda \\ M(t)=e^{m(e^t-1)} \\ Y = X_1 +\cdots X_n \sim Poisson(\lambda_1+\cdots \lambda_n)$$

$$E(X)=\sum^n_{x=0} x\dfrac{\lambda^x e^{-\lambda}}{x!} = \lambda$$

$$\begin{split} M_X(t) &= E(e^{tX}) \\ & = \sum^{\infty}_{x=0} e^{tx} \times \dfrac{\lambda^x e^{-x}}{x!} \\ &= e^{-\lambda} \sum^{\infty}_{x=0}\dfrac{(\lambda e^t)^x}{x!} \\ & = e^{\lambda (e^t-1)} \end{split}$$

Poisson & Binomial

In a rare event situation(n is so large and p is so small in binomial distribution), binomial distribution asymptotically can be a poisson distribution.

$$\lim_{n \to \infty}\dfrac{n!}{n^x(n-x)!}=1, \; \lim_{n \to \infty}(1-\frac{u}{n})^n=e^{-u}, \; \lim_{n \to \infty}(1-\frac{u}{n})^{-x}=1$$

$$\begin{split} \lim_{n \to \infty}b(x;n,p) &= \lim_{n \to \infty}\dfrac{n!}{x!(n-x)!}p^x(1-p)^{n-x} \\ &= \lim_{n \to \infty} \dfrac{n^x}{x!} \dfrac{n!}{n^x(n-x)!} p^x(1-p)^{n-x} \\ &= \lim_{n \to \infty} \dfrac{(np)^x(1-p)^{n-x}}{x!} \\ &= \lim_{n \to \infty} \dfrac{(np)^x(1-\frac{np}{n})^{n-x}}{x!} \\ &= \dfrac{\lambda^xe^{-\lambda}}{x!} \end{split}$$

Exercises