# Binomial

\defl{Binomial Distribution has the following properties:}

1. There are a fixed number of trials or observations, $n$, determined in advance.
2. Each trial can take on one of two possible outcomes, labeled ”success” and ”failure”.
3. Each trial’s outcome is determined independently of all the other trials.
4. The probability of a success and that of a failure remains the same from one trial to the next, and is denoted by $\pi$ and $1-\pi$, respectively.

\defm{Binomial Distribution:}
$P(X=x)=f(x)={n \choose x}\pi^{x}(1-\pi)^{n-x}$

• $X$ is Binomially distributed
• $x$ the number of successes, where $x = 0,1,\ldots,n$
• $n$ the number of trials
• $\pi$ the probability of success
• $1-\pi$ the probability of failure
• $\mu = E(X) = n\pi$
• $\sigma^2 = V(X)= n\pi(1-\pi)$

\defs{Examples}

• The number of heads out of 2 coin tosses.
• Assume that the probability of SET, Securities Exchange of Thailand, will end the day on the positive side is constant and each day’s outcome, positive or not, is independent of what occurred on prior days. The number of times the SET increases in 2 weeks, 10 working days.
• The number of women in a group of 20 randomly selected people.