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 :-
Uniform
- 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 λ (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: 𝜆e to the power -𝜆𝓍 , where lambda >0
- CDF: 1 -e to the power -𝜆𝓍
- Mean: 1/𝜆
- Variance: 1/𝜆 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.
Example:
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 :-
Uniform
- 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 λ (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: 𝜆e to the power -𝜆𝓍 , where lambda >0
- CDF: 1 -e to the power -𝜆𝓍
- Mean: 1/𝜆
- Variance: 1/𝜆 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.
Example:
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