Free Probability Distributions MCQs with Answers

• A. q
• B. p
• C. 0
• D. 1

Explanation: The binomial mean is E(X) = np. When n = 1, E(X) = 1 × p = p, which is also the mean of a Bernoulli variable.

• A. p= 0.5
• B. P 0.5
• C. p = 1
• D. None of these

Explanation: When p = 0.5, the binomial distribution is symmetric, so its mean and median coincide, and in the usual textbook treatment the mode is…

• A. p=0
• B. p>1/2
• C. p=1/2
• D. p=1/3

Explanation: A binomial distribution is negatively skewed when successes are more likely than failures, so p > 0.5.

• A. Three
• B. Two
• C. One
• D. Four
• E. None

Explanation: The binomial distribution has two parameters: n, the number of trials, and p, the probability of success.

• A. Less than mean
• B. Equal to mean
• C. Greater than mean
• D. Equal to standard deviation
• E. None of these

Explanation: For a binomial distribution, mean = np while variance = npq. Since q is less than 1 for a non-degenerate binomial distribution, npq is…

• A. p = q
• B. p > q
• C. p > q
• D. np > npq

Explanation: A binomial distribution is symmetrical when the success and failure probabilities are equal, so p = q = 0.5.

• A. Hpergeometric distribution
• B. Binomial distribution
• C. Geometric distribution
• D. None of these

Explanation: In a binomial model, each trial has the same success probability because trials are independent, as when sampling with replacement.

• A. Only 2 possible outcomes
• B. Have constant probability
• C. Must have at least 3 trials
• D. Trials must be independent

Explanation: A binomial distribution does not require at least three trials; it can be defined for any positive integer n, including one trial.

• A. 60
• B. 12
• C. 4.8
• D. 0

Explanation: The binomial variance is npq, where q = 1 - p = 2/5. Thus, variance = 20 × 3/5 × 2/5 = 4.8.

• A. n p
• B. np (1 - p)
• C. np/q
• D. nq/p

Explanation: For a binomial distribution, q represents the failure probability and equals 1 - p. The variance is therefore npq = np(1 - p).

• A. 0
• B. 2
• C. 1
• D. None of these

Explanation: In a binomial distribution, p is the success probability and q is the failure probability, with q = 1 - p. Therefore, p + q = 1.

• A. Positively skewed
• B. Negative skewed
• C. May be positive skewed or negative skewed.
• D. None of these

Explanation: The binomial skewness has the sign of q - p. It is positively skewed when p < q and negatively skewed when p > q, so either direction is…

• A. p > q
• B. p < q
• C. p = q
• D. both (a) and (b

Explanation: For a binomial distribution, positive skewness occurs when p is smaller than q, or p < 0.5.

• A. 5
• B. 96
• C. 5/96
• D. 96/5

Explanation: N denotes the number of repetitions of the experiment. Rolling five dice 96 times means the experiment is repeated 96 times, so N = 96.

• A. N.{nx}px qn-x
• B. N p q
• C. N.{n/x}px qn-x
• D. N.{n/x}p qn-x

Explanation: The binomial frequency for x successes is N multiplied by the binomial probability, namely N·C(n,x)·p^x·q^(n−x).

• A. n must assume a number between 1 and 20 or 25
• B. p must be a multiple of 10
• C. There must be at least 3 possible outcomes
• D. None of these

Explanation: A binomial experiment requires a fixed number of trials, two possible outcomes per trial, and constant success probability, but n is not…

• A. p < q
• B. q/p
• C. p > q
• D. p > 1/2

Explanation: Positive skewness in a binomial distribution occurs when failures are more probable than successes, so q > p or p < q.

• A. Constant Probability of Success
• B. Only two possible outcomes
• C. A fixed number of trials
• D. Equally likely outcomes

Explanation: The two outcomes need not be equally likely; they may have probabilities p and q with p different from q.

• A. Unequal
• B. Wrong
• C. Fair
• D. None of these

Explanation: For a binomial distribution, mean = np and variance = npq, so variance divided by mean must equal q.

• A. 1/q
• B. 1
• C. q
• D. 2q

Explanation: Symmetry occurs when the success and failure probabilities are equal, p = q.