Discrete Random Variables

• Random variables are numerical values from random experiments, used to model uncertain events like component lifetimes or exam scores.
• Discrete random variables have distinct values from a finite or countably infinite set, such as die faces or number of children.
• The Mass Probability Function describes the distribution of possible values and their probabilities.
• Mathematical tools like the Distribution Function, Expectation, Variance, Standard Deviation, and the Moment Generating Function are used in studying discrete random variables.
• Common discrete probability distributions include Bernoulli, Binomial, Poisson, and Geometric distributions.

Random Variables

• A random variable X on a space Ω is a function from Ω to real numbers, where for each interval I of real numbers, the set of all outcomes w in Ω such that X(w) is in I is an event in Ω.
• A discrete set is a finite or countably infinite set.
• A discrete random variable X on a fundamental space Ω is a real-valued function on Ω with values in a discrete set X, which is a subset of real numbers.

Discrete Random Variable Notation

• Notation:
  • X represents a set of possible values {xᵢ} where i belongs to an index set I.
  • P(X = a) denotes the probability that X takes the value a.
  • P(a ≤ X ≤ b) is the probability that X falls within the interval [a, b].
• More precisely:
  • P(X = a) = P({ω ∈ Ω : X(ω) = a})
  • P(a ≤ X ≤ b) = P({ω ∈ Ω : a ≤ X(ω) ≤ b})
• Similar meanings apply to symbols like P(X > a), P(X ≥ a), P(X ≤ a), P(X < a), P(a < X < b), P(a ≤ X < b), P(a < X < b).

Probability Mass Function

• The Probability Mass Function (PMF) or Probability Distribution of a discrete random variable X is a sequence of probabilities pₓ(k) = P(X = k) for all k in the set of possible values of X.
• Properties:
  • For all k in X(Ω), 0 ≤ pₓ(k) ≤ 1.
  • The sum of pₓ(k) over all k in X(Ω) equals 1.

Cumulative Distribution Function

• The Cumulative Distribution Function (CDF) of X, denoted Fₓ, is defined as Fₓ(x) = P(X ≤ x), which is the sum of pₓ(xᵢ) for all xᵢ less than or equal to x.
• Properties:
  • It is an increasing function.
  • Fₓ(-∞) = lim (x→-∞) Fₓ(x) = 0
  • Fₓ(+∞) = lim (x→+∞) Fₓ(x) = 1
• The cumulative distribution function of X depends only on the probability mass function of X; conversely, it characterizes this probability mass function.

Function of a Discrete Random Variable

• Given a random variable X with values in a finite set and a function g from X to another finite set Y, g(X) is a random variable with values in Y. Its probability distribution is P(g(X) = y) = Σ pₓ(x), where the sum is over all x in the preimage of y under g.

Expectation of a Discrete Random Variable

• The Expectation or mean is a measure of the central tendency of a random variable.
• Given a discrete random variable X with values {x₁, x₂,..., xₙ}, the expectation of X, denoted by E(X), is calculated as: E(X) = x₁ × pₓ(x₁) + x₂ × pₓ(x₂) + ... + xₙ × pₓ(xₙ) or E(X) = Σ xᵢpₓ(xᵢ), where the sum is taken over all possible values xᵢ of X.

Expectation of a Discrete Random Variable continued

• For a countable set of values X, the expectation is E(X) = Σ xᵢpₓ(xᵢ), where the sum is taken over all possible values xᵢ of X. The series must be absolutely convergent, i.e. Σ |xᵢ|pₓ(xᵢ) < +∞. Otherwise, the expectation is undefined.
• E(X) represents the weighted average of values X can take, weighted by their probabilities, indicating the value expected per repetition on average over many trials.

Common Discrete Distributions

• Discrete Uniform Distribution:
  • Applies to equiprobable events.
  • A random variable X with n possible values has a discrete uniform distribution if P(X=k) = 1/n for all k from 1 to n.
• Bernoulli Distribution:
  • Models an experiment with two outcomes: success (value 1) and failure (value 0).
  • Defined by a parameter p in [0,1], where P(X=1) = p and P(X=0) = 1-p.
• Binomial Distribution:
  • Represents the probability distribution of a random variable equal to the number of successes in n independent Bernoulli trials, each with probability p of success.
• P(X=k) = C(n,k) * pᵏ * (1-p)^(n-k)*
• Geometric Distribution:
  • Models the number of Bernoulli trials needed to get the first success.
  • P(X=k) = q^(k-1) * p where q = 1 - p.
  • Memoryless property: the conditional probability of needing k more trials given that n trials have already failed is the same as the original geometric distribution.
• Negative Binomial Distribution:
  • Distribution consists of considering a sequence of Bernoulli trials and defining the r.v. X as the number of trials to be carried out to obtain the r-th success

Common Discrete Distributions continued

• Hypergeometric Distribution:
  • An urn contains d white balls and N – d black balls, you draw n balls without replacement
• P(X = k) = (C(k, d) * C(n-k, N-d)) / C(n, N)*
• Poisson distribution:
  • Variable follows a Poisson distribution with parameter λ > 0 if X = N
• e^(-λ) * λ^(k) / k!*

Poisson Distribution related to Binomial Distribution

• The Poisson distribution can be approximated using the binomial distribution when n is large and p is small, by setting λ = np.

Expectation of a function of a discrete r.v.

• It is a discrete r.v. and (pₓ(xᵢ))ᵢ∈ₓ its probability distribution, construct Y = g(X) .

Theorem

• E[g(X)] = Σ g(xᵢ)pₓ(xᵢ)

Moment of a discrete r.v.

• E (Xᵏ) is called the moment of order k of the r.v. X

Variance of a discrete r.v.

• V(X) = E [(X – E(X))²]

Moment Generating Function

• Moment generating function is: Mx(t) = Gx(et) = E (et एक्स)