Probability Distribution on Amit Rajan

Commonly used Distributions (Part 2)

Fri, 16 Nov 2018 01:00:00 +0530

#### The Normal Distribution :

The normal distribution is a continuous distribution with any mean and a positive variance. The probability density function of a normal distribution with mean \(\mu\) and variance \(\sigma^2\) is given as:

\[f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]

The above normal distribution is denoted as \(X \sim N(\mu, \sigma^2)\) with mean and variance as:

\[\mu_X = \mu\]\[\sigma_X^2 = \sigma^2\]

For a normal distribution, about 68% of the population is in the interval \(\mu \pm \sigma\), 95% in \(\mu \pm 2\sigma\) and 99.7% in \(\mu \pm 3\sigma\). It is a widespread practice to convert the units of the normal population to the standard units which tells us that how many standard deviations an observation is from the population mean. It is sometimes called as z-score and is given as:

Commonly used Distributions (Part 1)

Thu, 15 Nov 2018 12:03:41 +0100

#### The Bernoulli Distribution :

Bernoulli trial is an experiment that can result in two outcomes: success (with probability \(p\)) and failure (with probability \(1-p\)). A Bernoulli random variable \(X\) can be represented as \(X \sim Bernoulli(p)\). It’s mean \(\mu_X\) and variance \(\sigma_X^2\) can be computed as:

\[\mu_X = 0 \times (1-p) + 1 \times p = p\]\[\sigma_X^2 = (0-p)^2(1-p) + (1-p)^2p = p(1-p)\]
#### The Binomial Distribution :

When a set of \(n\) independent Bernoulli trials are conducted, each with a success probability of \(p\), a random variable \(X\) which is equal to the number of success in these trials is said to have the binomial distribution with parameters \(n\) and \(p\) and is represented as \(X \sim Bin(n, p)\). Probability mass function of a binomial distribution can be computed as: