Probability and Statistics Quiz

Which of the following statements about the PDF of a continuous random variable is true?

• It gives the probability that the random variable takes any particular value.

• It must be non-negative everywhere and integrates to 1 over all possible values.

• It always has a finite range.

• It represents the sum of probabilities across all possible values of the random variable.

For a given random variable X with E(X) = 10 and Var(X) = 4, what is the standard deviation of X?

• 2

• 4

• 16

• 8

Which of the following correctly describes the relationship between the PMF and CDF of a discrete random variable?

• The PMF is the derivative of the CDF.

• The PMF is the integral of the CDF.

• The CDF is the cumulative sum of the PMF values.

• The PMF and CDF are unrelated.

If you have a bag containing 5 red balls and 7 blue balls, and you draw 3 balls without replacement, which probability distribution should you use to find the probability of drawing exactly 2 red balls?

• Binomial distribution

• Poisson distribution

• Hypergeometric distribution

• Normal distribution

Which of the following is a necessary condition for two events A and B to be independent?

• P(A) = P(B)

• P(A and B) = P(A) + P(B)

• P(A or B) = P(A) × P(B)

• P(A and B) = P(A) × P(B)

In a binomial experiment with n trials and success probability p, what is the expected value of the number of successes?

• n * p * (1 - p)

• n * p

• np / (1 - p)

• n * (1 - p)

If a random variable X follows a hypergeometric distribution with population size N, number of success states K, sample size n, and X representing the number of successes in the sample, which of the following is always true?

• E(X) = K / N

• E(X) = n * (K / N)

• E(X) = K

• E(X) = n + K

Which of the following represents the definition of a "counting" principle?

• Multiplying the probabilities of independent events

• Determining the number of ways to arrange or select items

• Calculating the area under the PDF curve

• Summing the probabilities of mutually exclusive events

A dealer in Las Vegas selects 10 cards from a standard deck of 52 cards.  Let Y be the number of diamonds in the 10 cards selected.  Which of the following best describes this setting?

• Y has a binomial distribution with n = 10 observations and probability of success p = 0.25.
• Y has a binomial distribution with n = 10 observations and probability of success p = 0.25, provided the deck is shuffled well.
• Y is a binomial distribution with n = 10 observations and probability of success p = 0.25, provided that after selecting a card it is replaced in the deck and the deck is shuffled well before the next card is selected.
• Y has a geometric distribution with n = 10 observations and probability of success p = 0.25