Probability Distributions 3D

Question 1

What parameter controls the spread of a Gaussian distribution?

Accepted Answer

The distribution's variance value — Variance (or standard deviation) determines how spread out the distribution is. Higher variance means a wider bell curve.

Question 2

What does a correlation of 0 imply between two variables?

Accepted Answer

No linear relationship — A correlation of 0 indicates no linear relationship between the variables, meaning one does not predict the other linearly.

Question 3

What does a probability distribution describe?

Accepted Answer

How probability spreads over the values — A distribution assigns probabilities (or densities) to all possible values a random variable can take, summarizing uncertainty.

Question 4

The key difference between a PMF and a PDF is that a PMF is for…

Accepted Answer

Discrete variables — A probability MASS function gives P(X=x) for discrete outcomes; a probability DENSITY function gives relative likelihood whose integral over an interval is a probability.

Question 5

The expected value E[X] of a random variable is…

Accepted Answer

Its probability-weighted average outcome — E[X] = Σ x·P(x) (or ∫ x·f(x)dx) — the mean you'd expect averaging infinitely many samples.

Question 6

Variance measures…

Accepted Answer

The spread around the mean — Variance quantifies how far values typically fall from the mean; standard deviation is its square root, in the original units.

Question 7

The Gaussian (normal) distribution is fully described by…

Accepted Answer

Its mean μ and variance σ² values — The bell curve N(μ,σ²) is parameterized by mean (center) and variance (width). It appears everywhere via the Central Limit Theorem.

Question 8

The Central Limit Theorem states that the sum/mean of many independent random variables tends toward a…

Accepted Answer

A normal distribution regardless of the original shape — The CLT explains why Gaussian noise is ubiquitous: averaging many independent effects produces an approximately normal distribution.

Question 9

A Bernoulli distribution models…

Accepted Answer

A single, plain binary trial — Bernoulli(p) is one yes/no event. The binomial distribution sums n independent Bernoulli trials.

Question 10

The Poisson distribution models…

Accepted Answer

Event counts given a rate λ — Poisson(λ) describes rare-event counts (e.g. arrivals per minute) with mean and variance both equal to λ.

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What parameter controls the spread of a Gaussian distribution?

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