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

f(x) = e^(−x)

Answer

f(x) = 1/(1 + x²)

Answer

f(x) = e^(−x²)

Question 2

A Type I error in hypothesis testing refers to:

Accepted Answer

Rejecting a true null hypothesis

Answer

Failing to reject a false null hypothesis

Answer

Accepting a false alternative hypothesis

Answer

Accepting a true null hypothesis

Question 3

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

Accepted Answer

f(x) = F'(x)

Answer

f(x) = F(x²)

Answer

f(x) = ∫F(x) dx

Answer

f(x) = 1 − F(x)

Question 4

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

Accepted Answer

Mean = λ, Variance = λ

Answer

Mean = λ, Variance = λ²

Answer

Mean = √λ, Variance = λ

Answer

Mean = λ², Variance = λ

Question 5

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

Accepted Answer

X and Y are linearly uncorrelated

Answer

X and Y are independent

Answer

X = Y

Answer

Var(X) = Var(Y)

Question 6

A confidence interval for a population mean is constructed so that:

Accepted Answer

The procedure captures the true mean in 95% of repeated samples

Answer

Exactly 95% of individual data points lie within it

Answer

The population mean is always at the center

Answer

The sample mean is 95% likely to equal the population mean