Probability
Discrete Random Variables
A random variable is a number whose value depends upon the outcome of a random experiement. Such as tossing a coin 10 times and let X be the number of Head.
A discrete random variable X has finitely countable values $x_i = 1, 2…$ and $p(x_i) = P(X = x_i)$ is called probability mass function.
Probability mass functions has following properties:
- For all i, $p(x_i) > 0$
- For any interval $P(X \in B) = \sum_{x_i \in B}p(x_i)$
- $\sum_{i}p(x_i) = 1$
There are many types of discrete random variable
- Unifrom discrete random variable
- $\mathbb{E} = \frac{x_1 + x2+ …}{n}$
- Bernoulli Random Variable
- $\mathbb{E} = p$
- Binomial random variable
- $\mathbb{E} = np$
- Probability mass function $P(X = i) = C(n, i)p^i(1-p)^{n - i}$
- Poisson random variable
- $P(x = i) = \frac{\lambda^i}{i!}e^{-\lambda}$
- $\mathbb{E} = \lambda$
- Gemometric random variable
- $P(X = n) = p(1 - p)^{n - 1}, where n = 1, 2, …$
- $\mathbb{E} = \frac{1}{p}$
Coninuous Random Variables
A random variable X is continuous if there exists a nonnegative function f so that, for every interval B, $$ P(X \in B) = \int_{B}f(x)dx $$
and f(x) is called the density of X. Also, it must hold that$\int_{-\infty}^{\infty}f(x)dx = 1$.
The function F = F(X) given by $$ F(x) = P(X <= x) = \int_{-\infty}^{x}f(s)ds $$
is called the distribution function of X and $F’(x) = f(x)$.