BMath Bachelor of Mathematics Probability and Statistics 1 — Questions and Answers
Question 1: If P(A) = 0.4, P(B) = 0.5, and A and B are independent, what is P(A ∩ B)?
- 0.9
- 0.2 (Correct answer)
- 0.1
- 0.45
Correct answer: 0.2
For independent events, P(A ∩ B) = P(A)·P(B) = 0.4 × 0.5 = 0.2.
Question 2: What does the expected value E[X] of a discrete random variable X represent?
- The most likely outcome
- The weighted average of all possible values (Correct answer)
- The median of the distribution
- The variance of X
Correct answer: The weighted average of all possible values
E[X] = Σ x·P(X = x) is the long-run average value of X over many repetitions of the experiment.
Question 3: Which distribution models the number of successes in n independent Bernoulli trials with probability p?
- Poisson distribution
- Normal distribution
- Binomial distribution (Correct answer)
- Exponential distribution
Correct answer: Binomial distribution
The Binomial distribution B(n, p) counts successes in n independent trials, each with success probability p.
Question 4: Bayes' Theorem states that P(A|B) equals:
- P(B|A) / P(A)
- P(A)·P(B|A) / P(B) (Correct answer)
- P(A ∩ B) / P(A)
- P(B) / P(A|B)
Correct answer: P(A)·P(B|A) / P(B)
Bayes' Theorem: P(A|B) = P(B|A)·P(A) / P(B), allowing us to update probabilities based on new evidence.
Question 5: The variance Var(X) of a random variable X is defined as:
- E[X²]
- E[X]²
- E[(X − E[X])²] (Correct answer)
- E[X] − E[X²]
Correct answer: E[(X − E[X])²]
Variance measures the expected squared deviation from the mean: Var(X) = E[(X − μ)²] = E[X²] − μ².
Question 6: The Central Limit Theorem states that as n → ∞, the sample mean X̄ of i.i.d. random variables approaches:
- The median of the population
- A normal distribution regardless of the population's distribution (Correct answer)
- The population standard deviation
- A uniform distribution
Correct answer: A normal distribution regardless of the population's distribution
The CLT guarantees that the sampling distribution of XÌ„ is approximately normal for large n, regardless of the underlying distribution.
BMath Bachelor of Mathematics Probability and Statistics 1
If P(A) = 0.4, P(B) = 0.5, and A and B are independent, what is P(A ∩ B)?