Topic 3 Probability Distributions Flashcards
What are the two main tools used to describe random variable distributions?
Distribution functions (CDF, PMF/PDF) and summary statistics (mean, variance)
The inverse of the CDF is called the ______ function.
quantile
What is a Bernoulli trial?
A random experiment with two possible outcomes (success/failure)
Give an example of a Bernoulli trial in civil engineering.
Whether a concrete batch passes or fails a quality test
Formula Recall
If P(X = k) = p^k (1 − p)^(1 − k) for k = 0 or 1
Then what is E(X) = __ ; Var(X) = __
E(X) = p ; Var(X) = p(1 − p)
X ~ B(1, p) represents a ________ distribution.
Bernoulli
Binomial
Formula Recall
P(X = k) =
nCk × p^k × (1 − p)^(n − k)
Binomial
Mean = np, Variance = ______
np(1 − p)
X ~ B(n, p) is a _______ distribution.
Binomial
Poisson Distribution
Flashcard 10: Formula
P(X = k) = (μ^k * e^(−μ)) / k!
E(X) = Var(X) = __
μ
What is the Poisson distribution often used to model?
Arrivals of independent events over time or space.
Geometric Distribution
Formula
P(X = k) =
p(1 − p)^k for k = 0, 1, 2, …
Geometric Distribution
E(X) = ___ ; Var(X) =____
(1 − p)/p
(1 − p)/p²
X ~ Geom(p) models the number of trials until the first ______.
success
Uniform Distribution
Flashcard 15: Formula
f(x) = ____
E(X) = ____
Var(X) = ____
1 / (b − a) for x ∈ [a, b]
(a + b)/2
(b − a)² / 12
X ~ U[a, b] means X is _________ distributed.
uniformly
Exponential Distribution
Flashcard 17: Formula
f(x) = ____
E(X) = ____
Var(X) = ____
λe^(−λx), x ≥ 0
1/λ
1/λ²
What kind of events does the exponential distribution model?
Time until next independent event (e.g., time until storm or failure)
Gamma Distribution
Formula
f(x) = ____
E(X) = ____
Var(X) = ____
(x^(⍺−1) * e^(−x/β)) / (β^⍺ Γ(⍺)) for x > 0
⍺β
⍺β²
X ~ Gamma(θ, k) generalises the ________ distribution.
exponential
Normal Distribution
Formula
f(x) = ____
E(X) = ____
Var(X) =_____
(1 / (σ√(2π))) * e^(−(x − μ)² / (2σ²))
μ
σ²
X ~ N(μ, σ²) means X is ______ distributed.
normally
Standard Normal Distribution
Formula
Z = ____
(X − μ)/σ
What’s the mean and variance of Z?
Mean = 0 ; Variance = 1
The standard normal CDF is denoted by __.
Φ(z)
Bernoulli is a Binomial with ___
Poisson is a limit of Binomial as ___
Exponential is a special case of ___
n = 1
n → ∞ and p → 0
Gamma (when k = 1)
What’s the difference between geometric and exponential distributions?
Geometric: discrete trials to first success.
Exponential: continuous time to event.
What is a confidence interval?
A range of values for a random variable X that covers p% of the PDF
For 95% confidence: q₂.₅(X) to _____.
q₉₇.₅(X)
Definition of Geometric Distribution
The Geometric Distribution gives the probability of
the number of independent k Bernoulli trials necessary
before achieving a “success”
Γ(⍺) =
∫ between ∞ and 0 for (x^(⍺−1) * e^(−x) dx
