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)

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
• Refer Here for the excel sheet used.

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