BMath Bachelor of Mathematics Probability and Statistics 3

Question 1

The law of total probability states that P(B) equals:

Accepted Answer

P(B|A)·P(A) + P(B|Aᶜ)·P(Aᶜ)

Answer

For a partition {A, Aᶜ}, P(B) = P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ), summing over mutually exclusive cases.

Question 2

For the exponential distribution with rate λ, the mean is:

Accepted Answer

1/λ

Answer

The exponential distribution Exp(λ) has mean 1/λ and variance 1/λ², modeling waiting times between Poisson events.

Question 3

In simple linear regression y = β₀ + β₁x + ε, what does β₁ represent?

Accepted Answer

The expected change in y for a one-unit increase in x

Answer

β₁ is the slope coefficient: it gives the expected change in the response variable y for each one-unit increase in the predictor x.

Question 4

The p-value in a hypothesis test represents:

Accepted Answer

The probability of observing a result as extreme as the data, assuming H₀ is true

Answer

The p-value measures how compatible the observed data are with H₀; a small p-value suggests evidence against H₀.

Question 5

For two independent random variables X and Y, Var(X + Y) equals:

Accepted Answer

Var(X) + Var(Y)

Answer

Independence implies Cov(X,Y) = 0, so Var(X + Y) = Var(X) + Var(Y).

BMath Bachelor of Mathematics Practice Test