BMath Bachelor of Mathematics Probability and Statistics 2

Question 1

The probability density function of the standard normal distribution N(0,1) is:

Accepted Answer

f(x) = (1/√(2π)) e^(−x²/2)

Answer

The standard normal PDF is φ(x) = (1/√(2π)) e^(−x²/2), symmetric about 0 with unit variance.

Question 2

A Type I error in hypothesis testing refers to:

Accepted Answer

Rejecting a true null hypothesis

Answer

A Type I error (false positive) occurs when we reject H₀ even though it is actually true; its probability is the significance level α.

Question 3

For a continuous random variable X with CDF F(x), the PDF f(x) satisfies:

Accepted Answer

f(x) = F'(x)

Answer

The PDF is the derivative of the CDF: f(x) = F'(x), which gives the density of probability at each point x.

Question 4

If X ~ Poisson(λ), what is both the mean and variance of X?

Accepted Answer

Mean = λ, Variance = λ

Answer

A key property of the Poisson distribution is that both its mean and variance are equal to the parameter λ.

Question 5

The covariance Cov(X, Y) = 0 implies:

Accepted Answer

X and Y are linearly uncorrelated

Answer

Zero covariance means no linear relationship between X and Y, but independence is a stronger condition that also rules out nonlinear relationships.

BMath Bachelor of Mathematics Practice Test