Discrete probability distributions describe the probability of outcomes for discrete random variables, which can take on a finite or countably infinite set of values. Here are the main components and properties:
Probability Mass Function (PMF)
The PMF P(X = x) gives the probability that a discrete random variable X takes a specific value x. It satisfies:
- 0 ≤ P(X = x) ≤ 1 for all x.
- ∑xP(X = x) = 1.
In quality control, a factory manager might use the PMF of the number of defective products in a batch to understand and predict production quality.
From a PMF, we can infer the probability of specific outcomes and the likelihood of various counts or occurrences. It helps in making informed decisions based on the distribution of discrete events.
Cumulative Distribution Function (CDF)
The CDF F(x) gives the probability that the random variable X is less than or equal to x:
F(x) = P(X ≤ x) = ∑t≤x P(X=t)
The CDF is a non-decreasing function and ranges from 0 to 1.
In risk assessment, a financial analyst might use the CDF to determine the probability that a portfolio’s return will be less than a certain threshold, helping to manage risk.
From a CDF, we can infer the cumulative probability of the random variable being less than or equal to a particular value. It provides an overall picture of the distribution and helps in understanding probabilities over intervals.
Moment Generating Function (MGF)
The MGF MX(t) is defined as the expected value of etX:
MX(t) = E[etX] = ∑xetx P(X = x)
The primary use of the MGF is to calculate the moments of a random variable, such as the mean (first moment), variance (second central moment), and higher-order moments. The moments can be derived by differentiating the MGF with respect to the parameter t and evaluating at t = 0.
1. Expected Value/ Mean (First Moment)
E[X]=∑xx⋅P(X = x)
2. Variance (Second Moment)
Var(X) = E[X2] − (E[X])2
The MGF, if it exists, uniquely determines the probability distribution of a random variable. This means that two random variables with the same MGF have the same probability distribution. This uniqueness property is useful for identifying the distribution of a random variable.
The MGFs simplify the calculations involving sums of independent random variables. If X1,X2,…,Xn are independent random variables with MGFs MX1(t),MX2(t),…,MXn(t), then the MGF of their sum Sn=X1+X2+…+Xn is the product of their individual MGFs:
MSn(t)=MX1(t)⋅MX2(t)⋅…⋅MXn(t).
This property is particularly useful in areas such as queueing theory, reliability engineering, and finance, where sums of random variables frequently occur.
