Source: Lecture 5 (9/15/22) | Course: Analytical Statistics (T), NYU
Tags: random variables, discrete distribution, PMF, probability mass function, probability distribution function, discrete uniform distribution, fair die, sample space
Difficulty: Introductory to Intermediate | Prerequisites: Basic probability concepts (sample spaces, counting rules, combinations)
Random variables are the bridge between raw probability experiments and the mathematics you will use for the rest of this course. Once you can assign a number to every outcome in a sample space, you unlock expectation, variance, and every distribution that follows. This lecture introduces the two broad families (discrete and continuous), then focuses on discrete random variables, their probability mass functions, and the most important discrete model: the binomial distribution. You should already be comfortable with sample spaces, basic counting rules, and the binomial coefficient before reading on.
A discrete random variable maps each outcome of an experiment to a number from a finite list. Its probability mass function (PMF) tells you the probability of each value, and those probabilities must sum to 1. The discrete uniform distribution is the simplest case (all values equally likely), while the binomial distribution models the count of successes in a fixed number of independent trials.
Random variable
A function that assigns a numerical value to each outcome in a sample space. Think of it as a rule that turns experimental results into numbers you can do maths with.
Discrete random variable
A random variable that can take on only a finite (or countably finite) number of distinct values, such as x₁, x₂, …, xₙ. In simple terms, you can list every possible value it might land on.
Probability distribution function (PDF) / Probability mass function (PMF)
The function fₓ(x) = P(X = x), which gives the probability that the random variable X equals a specific value x. Also referred to as the p.m.f. Think of it as a lookup table: plug in a value, get back its probability.
Discrete uniform distribution
A distribution where every possible value is equally likely. In simple terms, each outcome has exactly the same chance, like rolling a fair die.
Binomial coefficient
Written as C(N, n) or "N choose n", equal to N! / (n!(N−n)!). It counts the number of ways to choose n objects from a set of N, ignoring order. Think of it as the "how many different groups" calculator.
Random variables come in two flavours:
Discrete: X can take only a finite number of values (x₁, x₂, …, xₙ). This lecture's focus.
Continuous: X can take any value in an interval. Covered in later lectures.
For a discrete random variable X with possible values {x₁, x₂, …, xₙ}, the PMF fₓ(x) must satisfy three rules:
fₓ(x) = P(X = x) for each possible value
If x is not one of the possible values, then fₓ(x) = 0
The sum of all probabilities equals 1: Σ fₓ(xᵢ) = 1 (summed over i = 1 to n)
To find the probability that X falls in some set A (a subset of the real numbers), sum the PMF over every possible value inside A:
P(X ∈ A) = Σ fₓ(xᵢ) for all xᵢ in A
The simplest discrete distribution. If X is the result of rolling a fair die, each integer from 1 to 6 is equally likely.
The PMF is:
fₓ(x) = 1/k if x ∈ {x₁, x₂, …, xₖ}
fₓ(x) = 0 otherwise
This corresponds exactly to assigning simple (equal) probabilities across a sample space S = {x₁, x₂, …, xₖ}.
Define X = the number of heads observed when tossing 5 fair coins independently.
Setting up the sample space:
Total outcomes: n(S) = 2⁵ = 32
Number of ways to get exactly k heads: C(5, k), i.e. "5 choose k"
Calculating each probability:
P(X = 0) = C(5,0)/32 = 1/32
P(X = 1) = C(5,1)/32 = 5/32
P(X = 2) = C(5,2)/32 = 10/32
P(X = 3) = C(5,3)/32 = 10/32
P(X = 4) = C(5,4)/32 = 5/32
P(X = 5) = C(5,5)/32 = 1/32
These sum to 32/32 = 1, confirming a valid PMF. The shape of this distribution is symmetric and peaks at X = 2 and X = 3. This is, in fact, the PMF of a Binomial(5, 0.5) distribution.
PMF definition
fₓ(x) = P(X = x)
PMF validity conditions
fₓ(x) ≥ 0 for all x
fₓ(x) = 0 for any x outside the set of possible values
Σ fₓ(xᵢ) = 1
Probability of an event from the PMF
P(X ∈ A) = Σ fₓ(xᵢ), summed over all xᵢ in A
Discrete uniform PMF
fₓ(x) = 1/k for each of k equally likely values; 0 otherwise
Binomial coefficient
C(N, n) = N! / (n! × (N − n)!)
Discrete random variables and PMFs underpin quality control (counting defective items in a batch), polling (how many respondents favour a candidate), and network engineering (number of packets lost per second). Any time you are counting how many times something happens in a fixed setting, you are working with a discrete random variable.
Students often confuse the PMF with the CDF (cumulative distribution function). The PMF gives P(X = x), the probability of a single value. The CDF gives P(X ≤ x), the probability of being at or below a value. They are related but not the same thing.
A common mistake is forgetting that fₓ(x) = 0 for values outside the support set. If someone asks P(X = 3.5) for a fair die, the answer is 0, not "undefined".
Students sometimes assume "discrete" means "integer-valued". It does not. A discrete random variable just needs a finite (or countably finite) set of possible values. Those values could be 0.1, 0.5, and 0.9 if the experiment calls for it.
Forgetting to verify that probabilities sum to 1 is a frequent error when constructing a PMF from scratch.
⚠️ You will almost certainly be asked to verify that a given function is a valid PMF. Check: all values ≥ 0, and they sum to 1.
⚠️ Computing P(X ∈ A) by summing the relevant PMF values is a staple exam question. Be comfortable selecting the right values from the support set.
⚠️ Know how to use the binomial coefficient to count outcomes. Many discrete probability questions reduce to choosing and counting.
⚠️ The coin-toss example (5 fair coins) is a classic. Be able to reproduce the full PMF table from scratch.
True or False: A discrete random variable can take infinitely many values. (False, it takes a finite number of values.)
Fill in the blank: The sum of all PMF values must equal ___. (1)
True or False: If you roll a fair die, P(X = 4) = 1/6. (True)
Fill in the blank: C(5, 2) = ___. (10)
True or False: fₓ(x) can be negative for some values of x. (False, probabilities are always ≥ 0.)
Q: A discrete random variable X has possible values {1, 2, 3} with P(X=1) = 0.2, P(X=2) = 0.5, P(X=3) = 0.3. What is P(X ≥ 2)?
A: P(X ≥ 2) = P(X=2) + P(X=3) = 0.5 + 0.3 = 0.8
Q: You roll a fair six-sided die. What is the probability of rolling an even number?
A: The even values are {2, 4, 6}. Since this is a discrete uniform distribution, P(even) = 3 × (1/6) = 1/2.
Q: How many ways can you choose 3 items from a set of 7?
A: C(7, 3) = 7! / (3! × 4!) = 35
Q: When tossing 5 fair coins, what is the probability of getting exactly 2 heads?
A: P(X=2) = C(5,2) / 2⁵ = 10/32 = 5/16
Q: A PMF is given as fₓ(1) = 0.3, fₓ(2) = 0.4, fₓ(3) = c. Find c.
A: Since probabilities must sum to 1: 0.3 + 0.4 + c = 1, so c = 0.3.
This material leads directly into the binomial distribution (covered in the companion study notes), which is the most important discrete distribution for this course. The PMF concept extends to continuous random variables via the probability density function (PDF), which you will meet in later lectures. The binomial coefficient reappears when studying combinations, permutations, and the hypergeometric distribution.
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