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ML Foundations: Probability and Statistics

Foundational quiz on expectation, variance, covariance, Gaussian distributions, conditional probability, Bayes' rule, and entropy.

Learning path

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0 of 4 sections marked complete · about 25 minutes

Learning objectives

What you will be able to explain

  • What does the expectation of a random variable represent?
  • Compute a mean
  • Compute a population variance
  • What does positive covariance suggest?
  • Probability facts: true or false
  • Which parameters define a Gaussian distribution?

Section 01

What does the expectation of a random variable represent? to Compute a population variance

01

What does the expectation of a random variable represent?

Expectation is the probability-weighted average value. It is the basic notion of average behavior in probability and statistics.

Guided checkpoint

Choose the best interpretation.

02

Compute a mean

The mean is (2+4+6+8)/4=20/4=5(2 + 4 + 6 + 8) / 4 = 20 / 4 = 5.

Guided checkpoint

Compute the mean of the values 2,4,6,82, 4, 6, 8.

03

Compute a population variance

The mean is 33. The squared deviations are 4,0,44, 0, 4, whose average is 8/3≈2.6678/3 \approx 2.667.

Guided checkpoint

Compute the population variance of the values 1,3,51, 3, 5.

Section 02

What does positive covariance suggest? to Which parameters define a Gaussian distribution?

01

What does positive covariance suggest?

Positive covariance means the variables tend to move in the same direction relative to their means. It does not automatically imply a perfect or causal relationship.

Guided checkpoint

Choose the best interpretation.

02

Probability facts: true or false

These are basic statistical facts. Zero covariance means no linear relationship, but nonlinear dependence can still exist.

Guided checkpoint

Mark each statement as true or false.

03

Which parameters define a Gaussian distribution?

A Gaussian is fully specified by its location and spread. In one dimension that means mean and variance; in multiple dimensions it is mean vector and covariance matrix.

Guided checkpoint

Choose the correct answer.

Section 03

Compute a conditional probability to What does entropy measure?

01

Compute a conditional probability

By definition, P(A∣B)=P(A∩B)/P(B)=0.12/0.30=0.4P(A \mid B) = P(A \cap B) / P(B) = 0.12 / 0.30 = 0.4.

Guided checkpoint

If P(A∩B)=0.12P(A \cap B) = 0.12 and P(B)=0.30P(B) = 0.30, compute P(A∣B)P(A \mid B).

02

Which formula is Bayes' rule?

Bayes' rule reverses conditional probabilities by relating P(A∣B)P(A \mid B) to P(B∣A)P(B \mid A) and the marginal terms.

Guided checkpoint

Choose the correct identity.

03

What does entropy measure?

Entropy measures how spread out or uncertain a probability distribution is. It is central in information theory and classification losses.

Guided checkpoint

Choose the best interpretation.

Section 04

Why is cross-entropy widely used in classification?

01

Why is cross-entropy widely used in classification?

Cross-entropy is well suited to probabilistic classification because it strongly penalizes confident wrong predictions while rewarding well-calibrated correct ones.

Guided checkpoint

Choose the best answer.

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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