Source: Mid-Term Formula Sheet, STAT 601 (UCF)
Tags: probability distribution, expected value, binomial, Poisson, hypergeometric, normal distribution, exponential, uniform distribution, PDF, CDF, discrete, continuous, STAT 601
Difficulty: Intermediate Prerequisites: Chapters 1–3 (mean, variance, standard deviation), basic probability concepts, summation and factorial notation.
Chapters 1–3 described data you already had. Chapters 5 and 6 shift to modelling data you might see. A probability distribution is a mathematical function that assigns probabilities to every possible outcome of a random variable. Chapter 5 covers discrete distributions (countable outcomes: number of defects, number of successes), while Chapter 6 covers continuous distributions (any value on an interval: height, time, temperature). The specific distributions here, binomial, Poisson, normal, exponential, are workhorses that appear in nearly every applied statistics problem. They also underpin the sampling distributions and hypothesis tests in later chapters.
A probability distribution tells you the likelihood of each possible value a random variable can take. Discrete distributions (binomial, Poisson, hypergeometric) handle count-based outcomes. Continuous distributions (uniform, normal, exponential) handle measurements on a scale. The expected value and variance formulas for each distribution let you summarise their centre and spread without needing raw data.
Random variable
A variable whose value is determined by the outcome of a random process. Discrete if it takes countable values; continuous if it can take any value in an interval.
Probability mass function (PMF)
For discrete distributions, the function P(X = x) that gives the probability of each specific outcome. In simple terms, it answers "what is the chance of getting exactly this value?"
Probability density function (PDF)
For continuous distributions, f(x). The probability of a specific point is always zero; instead, you integrate f(x) over an interval to find the probability of falling in that range.
Cumulative distribution function (CDF)
F(x) = P(X ≤ x). The probability that the random variable takes a value less than or equal to x. Works for both discrete and continuous variables.
Expected value (mean)
The long-run average outcome. Discrete: μ = E(X) = Σ[xᵢ · P(xᵢ)]. Continuous: μ = ∫x · f(x) dx. Think of it as the weighted average of all possible outcomes, where the weights are their probabilities.
Variance of a discrete distribution
σₓ² = Σ{[xᵢ − E(X)]² · P(xᵢ)}. Alternatively: Var(X) = E(X²) − [E(X)]² = E(X²) − μ². This shortcut form is often faster in practice.
Covariance (discrete joint distribution)
σₓᵧ = Σ{[xᵢ − E(X)][yᵢ − E(Y)] · P(xᵢ, yᵢ)}. Captures how two discrete random variables move together.
Binomial distribution
Models the number of successes in n independent trials, each with success probability p. Think of it as "how many heads in n coin flips?"
Poisson distribution
Models the count of events in a fixed interval of time or space, given an average rate λ. Think of it as "how many customers arrive per hour?"
Hypergeometric distribution
Like the binomial, but for sampling without replacement from a finite population. Think of it as "drawing cards from a deck without putting them back."
Normal distribution
The bell curve. Defined by μ and σ. Symmetric, with tails extending to ±∞. Central to statistics because of the Central Limit Theorem.
Exponential distribution
Models the time between events in a Poisson process. If events arrive at rate λ per unit time, the waiting time between events follows an exponential with the same λ.
Continuous uniform distribution
Every value in [a, b] is equally likely. The simplest continuous distribution.
These rules hold regardless of the specific distribution:
E(X + Y) = E(X) + E(Y), always
Var(X + Y) = σ²ₓ + σ²ᵧ + 2σₓᵧ
If X and Y are independent, σₓᵧ = 0, so Var(X + Y) = σ²ₓ + σ²ᵧ
Portfolio return: E(P) = w · E(X) + (1 − w) · E(Y), where w is the proportion invested in asset X
Portfolio risk (standard deviation): σₚ = √[w²σ²ₓ + (1 − w)²σ²ᵧ + 2w(1 − w)σₓᵧ]
Diversification benefit comes from the covariance term: when σₓᵧ is negative, the portfolio risk can be lower than either individual asset's risk
Combinations (order does not matter): C(n, x) = n! / [x!(n − x)!]
Permutations (order matters): P(n, x) = n! / (n − x)!
Distribution | P(X = x) | Mean | Variance |
|---|---|---|---|
Discrete uniform | 1 / (b − a + 1) | (a + b) / 2 | [(b − a + 1)² − 1] / 12 |
Binomial | C(n,x) · pˣ · qⁿ⁻ˣ | np | npq |
Poisson | (λt)ˣ · e⁻ˡᵗ / x! | λt | λt |
Hypergeometric | C(s,x)·C(N−s, n−x) / C(N,n) | ns/N | (ns/N)·((N−s)/N)·((N−n)/(N−1)) |
Where q = 1 − p in the binomial. For Poisson, λ is the average rate and t is the interval length.
Distribution | PDF f(x) | CDF F(x) | Mean | Variance |
|---|---|---|---|---|
Continuous uniform | 1 / (b − a) | (x − a) / (b − a) | (a + b) / 2 | (b − a)² / 12 |
Normal | (1 / σ√2π) · e^[−½((x−μ)/σ)²] | N/A (use tables) | μ | σ² |
Exponential | λe⁻ˡˣ | 1 − e⁻ˡˣ | 1/λ | 1/λ² |
Symmetric about μ
Fully characterised by μ (centre) and σ (spread)
Standard normal: μ = 0, σ = 1. Convert any normal variable with z = (x − μ) / σ
Probabilities found via z-tables or software; no closed-form CDF
Memoryless property: P(X > s + t | X > s) = P(X > t). The probability of waiting another t units does not depend on how long you have already waited.
Mean waiting time = 1/λ
CDF gives the probability of the event occurring by time x: F(x) = 1 − e⁻ˡˣ
Discrete expected value: μ = E(X) = Σ[xᵢ · P(xᵢ)]
Discrete variance (shortcut): Var(X) = E(X²) − μ²
Continuous expected value: E(X) = ∫₋∞^∞ x · f(x) dx
Continuous variance: Var(X) = ∫₋∞^∞ (x − μ)² · f(x) dx
Variance of a sum: Var(X + Y) = σ²ₓ + σ²ᵧ + 2σₓᵧ
The Poisson distribution models events like call-centre arrivals, website hits per minute, or the number of typos per page. Insurance companies use it to model claim frequency. The exponential distribution then models the time between those same events, which is useful for planning staffing levels or server capacity.
The binomial distribution is the backbone of quality control. If a production line has a 2% defect rate and you inspect 50 items, the binomial tells you the probability of finding 0, 1, 2, … defects in that batch.
Students sometimes apply the binomial when sampling is done without replacement from a small population. If the sample is more than 5% of the population, use the hypergeometric instead (or apply a finite population correction).
The Poisson is sometimes confused with the binomial. The Poisson is for counts over a continuous interval (time, area) with a known average rate. The binomial is for a fixed number of discrete trials.
With continuous distributions, P(X = any specific value) = 0. Students sometimes try to compute P(X = 5) from a normal PDF; it is always zero. You need P(a ≤ X ≤ b).
The exponential distribution's mean is 1/λ, not λ. If events happen at a rate of 3 per hour, the mean waiting time is 1/3 of an hour, not 3 hours.
⚠️ Be able to identify which distribution fits a word problem: fixed trials with two outcomes = binomial; events per interval = Poisson; sampling without replacement from finite population = hypergeometric.
⚠️ Know the mean and variance formulas for each distribution by heart. The summary tables are exam gold.
⚠️ Var(X + Y) includes the 2σₓᵧ term. Only drop it when X and Y are independent.
⚠️ Portfolio risk is not simply the weighted average of individual risks, because of the covariance term.
⚠️ Exponential CDF: F(x) = 1 − e⁻ˡˣ. Quick to compute and commonly tested.
True or False: The binomial distribution requires that trials are independent.
Fill in the blank: If Poisson events occur at rate λ = 5 per hour, the mean of the distribution for a 2-hour window is ________.
True or False: For a continuous uniform distribution on [2, 10], the mean is 5.
Fill in the blank: The variance of the sum of two independent variables equals σ²ₓ ________ σ²ᵧ.
True or False: The exponential distribution is memoryless.
Answers: 1. True. 2. λt = 5 × 2 = 10. 3. False (mean = (2 + 10)/2 = 6). 4. + (plus; the covariance term drops out when independent). 5. True.
Q: A fair die is rolled 12 times. What distribution models the number of sixes, and what are its mean and variance?
A: Binomial with n = 12, p = 1/6. Mean = np = 2. Variance = npq = 12 × (1/6) × (5/6) = 10/6 ≈ 1.667.
Q: Calls arrive at a help desk at an average rate of 4 per hour. What is the probability of receiving 0 calls in a 30-minute window?
A: Poisson with λt = 4 × 0.5 = 2. P(X = 0) = (2⁰ · e⁻²) / 0! = e⁻² ≈ 0.1353.
Q: The time between arrivals at a coffee shop follows an exponential distribution with λ = 6 per hour. What is the probability the next customer arrives within 5 minutes?
A: Convert 5 minutes to hours: 5/60 = 1/12. F(1/12) = 1 − e⁻⁶ˣ⁽¹/¹²⁾ = 1 − e⁻⁰·⁵ ≈ 1 − 0.6065 = 0.3935.
Q: A portfolio is 60% in asset X and 40% in asset Y. E(X) = 8%, E(Y) = 12%. What is the portfolio's expected return?
A: E(P) = 0.6 × 8% + 0.4 × 12% = 4.8% + 4.8% = 9.6%.
The normal distribution here is the foundation for the sampling distribution of the mean (Chapter 7a) and for all z-based hypothesis tests (Chapter 9). The binomial links to the sampling distribution of proportions (Chapter 7b), where p = x/n is approximately normal under certain conditions. Portfolio risk from this chapter is a direct application of the variance-of-a-sum rule and reappears in finance courses.
probability distribution, PMF, PDF, CDF, expected value, discrete distribution, continuous distribution, binomial, Poisson, hypergeometric, normal distribution, bell curve, exponential distribution, uniform distribution, memoryless property, variance of a sum, portfolio return, portfolio risk, counting rules, combinations, permutations, STAT 601, UCF