Probability Theory: Univariate Models

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Questions and Answers

What does the binomial coefficient represent mathematically?

The number of ways to choose k items from N (correct)

The number of ways to arrange N items

The expected value in a binary distribution

The probability of success in a Bernoulli trial

Which distribution does the binomial distribution reduce to when N equals 1?

Poisson distribution

Geometric distribution

Bernoulli distribution (correct)

Normal distribution

In the logistic function, what does the output range from?

-1 to 1

0 to ∞

0 to 1 (correct)

-∞ to ∞

What role does the parameter ω play in the conditional probability distribution p(y|x, ω)?

It alters the probability calculation Signup and view all the answers

What is the Heaviside function primarily used to represent?

Thresholds in binary decisions Signup and view all the answers

What characteristic do all datasets in the Datasaurus Dozen share?

They all have the same low order summary statistics. Signup and view all the answers

Which visualization technique can better distinguish differences in 1d data distributions?

Violin plot Signup and view all the answers

What is a key limitation mentioned regarding the violin plot visualization?

It is limited to visualizing 1d data. Signup and view all the answers

Bayes' theorem is compared to which theorem in geometry?

Pythagoras's theorem Signup and view all the answers

In the context of Bayesian inference, what does the term 'inference' refer to?

Generalizing from sample data. Signup and view all the answers

What is the purpose of the simulated annealing approach as mentioned in the content?

To optimize the shape of datasets. Signup and view all the answers

What do the central shaded parts of the box plots indicate?

Median and inter-quartile range of the datasets. Signup and view all the answers

What kind of data is Bayes' rule primarily applied to?

Probabilistic data. Signup and view all the answers

What is a random variable?

An unknown quantity that can change and has different outcomes. Signup and view all the answers

Which of the following best describes a discrete random variable?

It can only take on a finite or countably infinite number of outcomes. Signup and view all the answers

What does the probability mass function (pmf) compute?

The probabilities of events corresponding to setting the rv to each possible value. Signup and view all the answers

Which of the following statements is true regarding the properties of the pmf?

All probabilities must be between 0 and 1, inclusive. Signup and view all the answers

In the context of rolling a dice, which of the following represents the sample space?

The set of numbers {1, 2, 3, 4, 5, 6}. Signup and view all the answers

What is an example of a degenerate distribution?

A distribution that assigns all probability mass to a single outcome. Signup and view all the answers

How is a continuous random variable defined?

It can take on any value within a given range of real numbers. Signup and view all the answers

What does the event of 'seeing an odd number' represent if X is the outcome of a dice roll?

X = {1, 3, 5} Signup and view all the answers

What does the variable $Y$ represent in the context of univariate Gaussians?

A mixture component indicator variable Signup and view all the answers

In the formulas provided, what does $V[X]$ represent?

The variance of the random variable $X$ Signup and view all the answers

Which of the following statements about Anscombe’s quartet is true?

All datasets in Anscombe's quartet have the same low order summary statistics. Signup and view all the answers

What do the terms $ heta_y$ and $ u_y$ likely refer to in the distribution $N(X| heta_y, u_y)$?

The variance and mean of the Gaussian respectively Signup and view all the answers

What does the notation $E[Y|X]$ represent?

The expected value of $Y$ given $X$ Signup and view all the answers

Which equation highlights the relationship between the variance and the expectations of random variables?

$V[X] = E[X^2] - (E[X])^2$ Signup and view all the answers

What is likely the role of the hidden indicator variable $Y$ in the mixture model?

To determine which mixture component generates the observation Signup and view all the answers

What can we infer if the datasets in Anscombe's quartet appear visually different?

They can produce different correlation results despite similar statistics. Signup and view all the answers

How can the joint distribution of two random variables be represented when both have finite cardinality?

As a 2D table where entries sum to one. Signup and view all the answers

What is the mathematical expression for obtaining the marginal distribution of variable X?

$p(X = x) = \sum p(X = x, Y = y)$ Signup and view all the answers

What does it mean if two random variables, X and Y, are independent?

Their joint distribution can be represented as the product of their marginal distributions. Signup and view all the answers

How is the conditional distribution of Y given X defined mathematically?

$p(Y = y|X = x) = \frac{p(X = x, Y = y)}{p(X = x)}$ Signup and view all the answers

What is the purpose of using the sum rule in probability?

To compute marginal distributions from joint distributions. Signup and view all the answers

Which of the following correctly summarizes the joint distribution in probabilistic terms?

$p(x, y) = p(x)p(y)$ when X and Y are independent. Signup and view all the answers

In the context of joint distributions, what does the term 'marginal' refer to?

The sum of all probabilities at the edge of a table. Signup and view all the answers

How can the joint distribution table be restructured if the variables are independent?

As two separate 1D vectors representing each variable. Signup and view all the answers

What is the output of the sigmoid function when applied to a > 0?

ϑ(a) Signup and view all the answers

How is the log-odds 'a' defined in relation to the probability 'p'?

a = log(p / (1 - p)) Signup and view all the answers

Which function maps the log-odds 'a' back to probability 'p'?

Sigmoid function Signup and view all the answers

In binary logistic regression, what form does the linear predictor take?

f(x; ω) = w^T x + b Signup and view all the answers

What does the function p(y = 1|x, ω) represent in the context of the sigmoid function?

The probability of y = 1 given x Signup and view all the answers

What is the output of the logit function when applied to probability 'p'?

log(p / (1 - p)) Signup and view all the answers

Which of the following correctly describes the inverse relationship between the sigmoid and logit functions?

The sigmoid function maps log-odds to probability, while the logit function does the opposite. Signup and view all the answers

What represents the probability distribution in binary logistic regression?

Bernoulli distribution Signup and view all the answers

Study Notes

Probability: Univariate Models

Probability theory is common sense reduced to calculation.
Two interpretations of probability exist: frequentist and Bayesian.
Frequentist interpretation: probability represents long-run frequencies of events.
Bayesian interpretation: probability quantifies uncertainty or ignorance about something.
Bayesian interpretation models uncertainty about one-off events.
Basic rules of probability theory remain consistent despite differing interpretations.
Uncertainty can stem from ignorance (model uncertainty) or intrinsic variability (data uncertainty).

Probability as an Extension of Logic

Probability extends Boolean logic.
An event (A) can either hold or not hold.
Pr(A) represents the probability of event A being true.
Values range from 0 to 1 (inclusive).
Pr(A) = 0 means event A will not happen, Pr(A) = 1 means event A will happen.

Probability of Events

Joint probability: Pr(A, B) or Pr(AB) is the probability of both A and B occurring.
If A and B are independent, then Pr(A, B) = Pr(A) x Pr(B).
Conditional probability: Pr(B|A) is the probability of B happening given A has occurred.
Pr(B|A) = Pr(A, B)/Pr(A)
Conditional independence: events A and B are conditionally independent given C if Pr(A, B|C)= Pr(A|C) x Pr(B|C).

Random Variables

Random variables (r.v.) are unknown or changeable quantities.
Sample space: the set of possible values of a random variable.
Events are subsets of outcomes in a given sample space,
Discrete random variables have finite or countably infinite sample spaces.
Continuous random variables take on any value within a given range.

Cumulative Distribution Function (CDF)

Cumulative distribution function (CDF) of a random variable X, denoted by P(x), is the probability that X takes on a value less than or equal to x.
P(x) = Pr(X ≤ x)
Pr (a ≤ X ≤ b) = P(b) – P(a)

Probability Density Function (PDF)

Probability density function is derived from the CDF.
PDF is the derivative of the CDF.
Pr (a ≤ X ≤ b) = integral of p(x) dx from a to b

Quantiles

Quantile function is the inverse of the CDF.
P-¹ (q) is the value x such that Pr (X ≤ xq) = q

Moments of a Distribution

Mean (μ): the expected value of a distribution.
E [X] = integral(x * p(x) dx) for continuous rv's.
E [X] = Σ (x * p(x)) for discrete rv's.
Variance (σ²): the expected squared deviation from the mean.
V [X] = E [(X - μ)²]
Standard Deviation (σ): the square root of the variance.
Mode: the value with the highest probability or probability density.

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Description

This quiz covers the fundamental concepts of probability theory with a focus on univariate models. It explores different interpretations of probability, such as frequentist and Bayesian, and discusses how probability extends Boolean logic. Test your knowledge on the principles and applications of probability.