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Random Variables and Probability Distribution

Random Variables:-

A variables whose value is subject to variations due to randomness.

The mathematical function describing this randomness (the probabilities for the set of possible values a random variable can take is called a probability distribution.)

Continuous and Discrete probability density functions.

 Continuous Distribution :-
Probability of certain height

Total Probability of all outcomes

Probability Density functions (PDFs) and Cumulative Density Functions (CDF)

Going from PDF to CDF and vice versa

Common distributions :-


  - Discrete
      The six sided dice, coin toss
       Formula for pdf: f (X = 𝒙) = 1/k for all 𝓍that belongs to a specific set with k elements And f (X = 𝓍) = 0 for all other values of x.

  - Continuous
       Number of seconds past the minute
       Exact age of a randomly selected person between the ages of 50-60
       Formula for PDF:
                    f (x) = 1/b-a for a ≤ x ≤ b  OR  0 for x < a and x > b
       What is the CDF, mean and Variance ?
     CD = 𝓍-a/b-a
      Mean = 1/2 (b+a)
      Variance = 1/2 (b-a)2

Binomial :-

  - What is it +Example: Toy problem
 - Example Real-world: Probability of 3 out of 10 mergers. Probability of there being 5 defective products in a batch of 20.
  - Formula for CDF is just the summation
  - It is more useful for small n's
  - Mean: np, variance: np(1-n)

Poisson :-

  - Discrete distribution that signifies the probability of 'x' occurrences of a certain event over a certain period of time or space.
  - Example: Number of defaults per month, Number of banks per square kilometer.
  - Mean and variance are Ξ» (lambda >0).

Geometric :-

  - Number of attempts before an event
  - The interarrival distribution counterpart of a binomial. The coin toss case (uniform, binomial, geometric)

  - Mean is 1/p , and variance 1-p/ p square .

Exponential :-

  - The interarrival times of the Poisson distribution
  - The continuous version of the geometric distribution
  - Memoryless
  -PDF: πœ†e to the power -πœ†π“ , where lambda >0
  - CDF: 1 -e to the power -πœ†π“
  - Mean: 1/πœ†
  - Variance: 1/πœ† square

Parallels to the Binomial, Exponential, Geometric

Working and Distributions

Normal :-

  - Bell shaped curve
  - Mean, variance, CDF
  - Height, weight, etc.
  - Many things after removal of outliers
  - Binomial Approximation
  - Central Limit Theorem (CLT)
  - Sampling distributions

Normal Distributions : Total Annual household income to explain outlier removal:

Binomial Approximation :-

Review of PDF, mean and variance
 - PDF n/k p to the power k (1-p) n-k
 - Mean = np
 - Variance = np (1-p)

Construct a normal distribution with the above mean and variance and use that to answer distribution related questions.

Central Limit Theorem :-

The aggregation of a sufficiently large number of independent random variables results in a random variable which will be approximately normal.


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