Discrete Random Variable

A random variable is discrete if the values it can take on within an interval is finite.

PMF and CDF

The probability mass function (PMF) describes the probability distribution over a discrete random variable.

$p(x)=P(X=x)$

The cumulative distribution function (CDF) specifies the probability of an observation being equal to or less than a given value.

$F(x)=P(X\leq x)$

We usually have tables for these in the case of discrete random variables.

Statistics

Expected value (mean):

$\mu =E(X)=\sum x_{i}P(X=x_{i})$

Distributions

Bernoulli

The bernoulli distribution describes the random variable of an experiment that has two outcomes and is performed once. The outcomes are either success or failure.

$X\sim Bernoulli(p)$

PMF

$p(1)=p,p(0)=1-p$

Statistics

$\mu =p$

$\sigma _{X}^{2}=p(1-p)$

Binomial

Repeating a bernoulli experiment $n$ times and we get a binomial random variable.

Consider an experiment with exactly two possible outcomes, conducted n times independently. The variable of interest $X$ is the number of successful trials. The distribution relies on the number of trials and the probability of success.

$X\sim Binomial(n,p)$

PMF

$(_{x}^{n})p^{x}(1-p)^{n-x}$

Statistics

$\mu =np$

$\sigma ^{2}=np(1-p)$

Poisson

The poisson distribution is used when we know the average rate of occurrence for a particular event over a particular time period.

• There must be fixed interval of the time or space
• Events happen with a known average rate independent of time or the last event.
• The average rate of occurrence per unit of time/sace is the rate parameter $\lambda$

Poisson distribution approximates binomial distribution when n is large and p is small, used to model rare events. Normally it is used to measure the number of events in a unit time, whereas exponential distribution models the amount of waiting time until an event.

I'm sleepy I'll write the details later... zzz...