**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:

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