Unit 4: Probability & Random Variables

Question 1

Complement Rule

Answer 1

P(Aᶜ) = 1 − P(A)

Question 2

Addition Rule

Answer 2

P(A ∪ B): Union
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Question 3

Multiplication Rule (independent)

Answer 3

P(A ∩ B): Joint probability
P(A ∩ B) = P(A) × P(B)

Question 4

Conditional Probability

Answer 4

P(A | B): A given B
P(A | B) = P(A ∩ B) / P(B)

Question 5

Expected Value (mean)

Answer 5

E(X): Expected value(Mean of a random variable)
E(X) = Σxᵢ × P(xᵢ)

Question 6

Variance of a Discrete RV

Answer 6

Var(X): Variance(Spread of a random variable)
xᵢ: specific outcome of the random variable X.
Var(X) = Σ(xᵢ − μ)² × P(xᵢ)

Question 7

Binomial Probability

Answer 7

P(X = k): probability of k successes in n trials
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ

Question 8

Mean and SD of Binomial

Answer 8

n: sample size
p: Population proportion
μ = n × p
σ = √[np × (1 − p)]