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Discrete Probability Distributions

This overview covers the essentials of discrete probability distributions, including random variables, probability mass functions (pmf), and cumulative distribution functions (cdf). A random variable transforms elements of a sample space into numerical values, while a discrete random variable has a finite or countable sample space. The properties of pmf, including non-negativity and normalization, are crucial, as well as the definition and calculation of mathematical expectation and variance. Understanding these concepts is fundamental for analyzing and modeling random phenomena.

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Discrete Probability Distributions

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  1. Discrete Probability Distributions • A sample space can be difficult to describe and work with if its elements are not numeric. • Random Variable • A random variable is a function that assigns each element in the sample space to a number. • The random variable X has range: • {x|x=X(s), for all s in S}. • More than one random variable can be associated with an experiment.

  2. Discrete Random Variable • A discrete random variable is a random variable that has a finite or countable sample space. • A sample space that can be mapped to the integers is said to be countably infinite (or countable).

  3. Probability Distribution • The probability distribution of a rv describes the distribution of total probability to all possible values of the rv. • A discrete probability distribution, called a probability mass function (pmf), specifies the probability of each distinct element in the sample space. p(x) = P( X = x ) = P(all s in S: X(s)=x)

  4. Properties of p(x) • p(x)≥0 for all x in S • ∑S p(x) = 1 • If A is a subset of S, then P(A) = ∑A p(x) Note: A pmf can be displayed nicely with a line graph or a probability histogram.

  5. Cumulative Distribution Function (cdf) • The cdf of a discrete random variable X with pmf p(x) is defined for each x as: F(x) = P( X ≤ x ) = ∑y:y≤x p(y) • For any number x, F(x) is the probability that the rv X will be at most x. • The graph of F(x) for a discrete rv is a step function. • For any two numbers a and b with a ≤ b, P( a ≤ x ≤ b ) = F(b) – F(a-) where a- represents the largest value of X less than a

  6. Mathematical Expectation • The mathematical expectation (expected value) of a discrete rv is the weighted average of all possible values of the rv, where the weight associated with each outcome is its probability. • The Expected Value of X • Let X be a discrete rv with pmf p(x). The expected value (or mean value) of X is: E[X] = µx = µ = ∑S x · p(x)

  7. Mathematical Expectation • The Expected Value of a Function of X • Let X be a discrete random variable with pmf p(x). The expected value of a function h(x) is E[h(X)] = µh(x)= ∑S h(x) · p(x) • Note that E[h(x)] only exists if ∑S h(x) · p(x) converges, therefore exists.

  8. Properties of E[X] • If c is a constant, then E[c] = c. • If c is a constant, then E[cX] = c·E[X]. • If c and d are constants, then E[cX+d] = c·E[X]+d. • If c is a constant and u(x) is a function, then E[c·u(X)] = c·E[u(X)].

  9. Properties of E[X] • Property of a Linear Operator • If ci are constants and ui(x) are functions, then E[c1·u1(X)+c2·u2(X)+…+cn·un(X)] = c1·E[u1(X)] + c2·E[u2(X)] + … + cn·E[un(X)] = ∑i=1,n ci · E[ui(X)]

  10. Variance of Discrete RV • Let X be a discrete random variable with pmf p(x). The Variance of X is • The variance of X measures the amount of spread in the distribution of X.

  11. Variance of a Discrete RV • Easier forms for the variance of X include:

  12. Standard Deviation • Let X be a discrete random variable with pmf p(x). The Standard Deviation of X is the square root of the Variance of X. • The standard deviation is commonly used as the measure of spread in a distribution

  13. Variance of a Function • Let X be a discrete rv with pmf p(x). The variance of a function h(X) is:

  14. Variance of a Function • If a and b are constants and Var(X)=σ2, then • Var( a·X ) = a2 · σ2 • Var( X+b ) = σ2 • Var( a·X+b ) = a2· σ2

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