Discrete Probability Distribution

A probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

Summary measures of a probability distribution are mean (central tendency), Variance (dispersion or spread) and Skewness or Kurtosis (shape).

$$X: {x_1, x_2 … x_n}$$ $$P: {p_1, p_2 … p_n}$$ $$Mean=E(X)= µ_x=\sum_{i=1}^n p_i x_i $$ $$Variance= Var(X)= σ_x^2=\sum_{i=1}^n p_i (x_i-µ_x)^2= \sum_{i=1}^n p_ix_i^2-µ_x^2 $$

Relationship between mean and variance for linear functions:

$$Y=α+βX$$ $$E(Y)= α+βE(X)$$ $$Var(Y)= β^2 Var(X)$$
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