BMath Bachelor of Mathematics Probability and Statistics 2 — Questions and Answers
Question 1: The probability density function of the standard normal distribution N(0,1) is:
- f(x) = e^(−x)
- f(x) = (1/√(2π)) e^(−x²/2) (Correct answer)
- f(x) = 1/(1 + x²)
- f(x) = e^(−x²)
Correct answer: f(x) = (1/√(2π)) e^(−x²/2)
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:
- Failing to reject a false null hypothesis
- Rejecting a true null hypothesis (Correct answer)
- Accepting a false alternative hypothesis
- Accepting a true null hypothesis
Correct answer: Rejecting a true null hypothesis
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:
- f(x) = F(x²)
- f(x) = F'(x) (Correct answer)
- f(x) = ∫F(x) dx
- f(x) = 1 − F(x)
Correct answer: f(x) = F'(x)
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?
- Mean = λ, Variance = λ²
- Mean = λ, Variance = λ (Correct answer)
- Mean = √λ, Variance = λ
- Mean = λ², Variance = λ
Correct answer: Mean = λ, Variance = λ
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:
- X and Y are independent
- X and Y are linearly uncorrelated (Correct answer)
- X = Y
- Var(X) = Var(Y)
Correct answer: X and Y are linearly uncorrelated
Zero covariance means no linear relationship between X and Y, but independence is a stronger condition that also rules out nonlinear relationships.
Question 6: A confidence interval for a population mean is constructed so that:
- Exactly 95% of individual data points lie within it
- The procedure captures the true mean in 95% of repeated samples (Correct answer)
- The population mean is always at the center
- The sample mean is 95% likely to equal the population mean
Correct answer: The procedure captures the true mean in 95% of repeated samples
A 95% CI means that 95% of intervals constructed this way from repeated samples will contain the true population mean.
BMath Bachelor of Mathematics Probability and Statistics 2
The probability density function of the standard normal distribution N(0,1) is: