Data Science Classroom Series – 04/Dec/2021

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 Preview
  • 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 Preview
    • No people are diabetic Preview
    • Exactly 2 are diabetic Preview
    • Atleast 2 are diabetic Preview
  • Excel =>
    • pmf = BINOM.DIST(x,n,p,FALSE)
    • cdf = BINOM.DIST(x,n,p,TRUE) Preview

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: Preview
  • 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 Preview
      • 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 Preview
  • Refer Here for the excel sheet used.

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