## Probability Distributions

- Probability Distribution is a mathematical function that gives probabilities of occurences of differnt possible outcomes
- Terminology
- Functions for discrete variables
- Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value
- Cumulative distribution function (cdf): Genereal term to indicate all various possible outcomes for discrete random variable

- Functions for Continuous Variables
- Probability density function (pdf): function whose value at any given sample (point) in the sample space.
- Cumulative distribution function (cdf)

- Functions for discrete variables

## Binomial Distribution

- This is distribution for discrete variables
- There are two possible outcomes per trail
- The probability of success(p) is same across all the trails
- The number of trails (n) is fixed
- Each trail is independent
- pmf formula
- Example: Studies show diabetics effects 9% of the population. A random sample of 10 people is taken
- Find the probability that All 10 people are diabetic
- No people are diabetic
- Exactly 2 are diabetic
- Atleast 2 are diabetic

- Excel =>
- pmf = BINOM.DIST(x,n,p,FALSE)
- cdf = BINOM.DIST(x,n,p,TRUE)

## Poisson Distrubution

- This is distribution for discrete variables
- This distribution can be used in scenarios that describe number of events occuring in a fixed interval or region of oppurtunity
- Requires only one parameter λ (mean)
- Assumptions:
- The rate at which events occur is constant
- The events are independent

- pmf:
- Example: Quality Thought has begun advertizing on youtube to direct the traffic to their websites, where students can enroll to courses. The number of click-through-sale from the ad is Poisson distributed with a mean of 12 click-through sales per day
- Find the probability of getting
- Exactly 10-click through sales in the first day
- Atleast 10 click through sales in the first days
- More than one sale in first one hours

`λ = 12 per day = 12/24 per hour = 0.5 P(X>=2) = 1-cdf(2) = 0.09`

- Excel =>
- pmf: POISSON.DIST(x,λ,FALSE)
- cdf: POISSON.DIST(x,λ,TRUE)

- The plots

- Find the probability of getting
- Refer Here for the excel sheet used.