Q.1 Which of the following is a discrete probability distribution?
Normal distribution
Poisson distribution
Exponential distribution
Uniform continuous distribution
Explanation - Poisson distribution is discrete because it deals with countable events occurring in a fixed interval.
Correct answer is: Poisson distribution
Q.2 The mean of a binomial distribution with parameters n and p is:
np
n(1-p)
√(npq)
n^2p
Explanation - For a binomial distribution, the expected value or mean is given by the product of the number of trials (n) and probability of success (p).
Correct answer is: np
Q.3 For a Poisson distribution with mean λ, the variance is:
λ
λ^2
√λ
1/λ
Explanation - In a Poisson distribution, the variance is equal to the mean λ.
Correct answer is: λ
Q.4 If X is a continuous random variable, the probability that X takes any exact value is:
1
0
Depends on X
Cannot be determined
Explanation - For continuous distributions, the probability of X taking any single exact value is 0 because there are infinitely many possible values.
Correct answer is: 0
Q.5 Which probability distribution is used to model the number of successes in n independent trials?
Normal
Binomial
Exponential
Poisson
Explanation - The binomial distribution counts the number of successes in a fixed number of independent Bernoulli trials.
Correct answer is: Binomial
Q.6 The standard deviation of a binomial distribution is given by:
√(npq)
npq
n√p
p/q
Explanation - The standard deviation of a binomial distribution is the square root of the product of number of trials, probability of success, and probability of failure.
Correct answer is: √(npq)
Q.7 Which of the following is a property of all probability distributions?
Sum of probabilities = 1
Mean = Variance
All outcomes are equally likely
Distribution is symmetric
Explanation - For any probability distribution, the total probability across all outcomes must equal 1.
Correct answer is: Sum of probabilities = 1
Q.8 For a continuous uniform distribution between a and b, the mean is:
(a+b)/2
a+b
√(ab)
(b-a)/2
Explanation - The mean of a continuous uniform distribution is the midpoint of the interval [a, b].
Correct answer is: (a+b)/2
Q.9 A distribution with a probability density function f(x) = λe^(-λx), x>0 is called:
Poisson distribution
Binomial distribution
Exponential distribution
Uniform distribution
Explanation - The exponential distribution is continuous and often models the time between events in a Poisson process.
Correct answer is: Exponential distribution
Q.10 If the probability of success in a Bernoulli trial is p, the probability of failure is:
1-p
p
√p
p/(1-p)
Explanation - In a Bernoulli trial, the probability of failure is the complement of the probability of success.
Correct answer is: 1-p
Q.11 Which of the following distributions is symmetric?
Poisson with small λ
Normal distribution
Exponential distribution
Geometric distribution
Explanation - The normal distribution is symmetric around its mean.
Correct answer is: Normal distribution
Q.12 The sum of two independent Poisson random variables with means λ1 and λ2 is:
Poisson with mean λ1+λ2
Normal with mean λ1+λ2
Binomial with parameters λ1+λ2
Exponential with rate λ1+λ2
Explanation - The sum of two independent Poisson random variables is also Poisson, with mean equal to the sum of individual means.
Correct answer is: Poisson with mean λ1+λ2
Q.13 For a geometric distribution with success probability p, the mean is:
1/p
p
q/p
p/q
Explanation - The expected number of trials to get the first success in a geometric distribution is 1/p.
Correct answer is: 1/p
Q.14 Which distribution is often used to model rare events over time?
Binomial
Poisson
Uniform
Normal
Explanation - The Poisson distribution models the number of rare events occurring in a fixed interval of time or space.
Correct answer is: Poisson
Q.15 The variance of a geometric distribution with success probability p is:
(1-p)/p^2
1/p
p(1-p)
p^2/(1-p)
Explanation - For a geometric distribution, the variance is calculated as (1-p)/p^2.
Correct answer is: (1-p)/p^2
Q.16 If X is normally distributed with mean μ and standard deviation σ, then P(μ-σ < X < μ+σ) is approximately:
68%
50%
95%
99%
Explanation - In a normal distribution, about 68% of the data lies within one standard deviation of the mean.
Correct answer is: 68%
Q.17 Which of the following is a characteristic of a Poisson distribution?
Mean equals variance
Always symmetric
Defined only for negative integers
Continuous random variable
Explanation - In a Poisson distribution, the mean and variance are equal.
Correct answer is: Mean equals variance
Q.18 For a continuous random variable, the total area under the probability density function curve is:
1
0
Depends on distribution
Cannot be determined
Explanation - The total probability for a continuous random variable must equal 1, represented by the area under its PDF curve.
Correct answer is: 1
Q.19 Which distribution is used for modeling time until the next event?
Binomial
Exponential
Poisson
Uniform
Explanation - The exponential distribution is used to model the waiting time until the next event occurs in a Poisson process.
Correct answer is: Exponential
Q.20 A probability distribution function must satisfy which condition?
f(x) ≥ 0 for all x
f(x) can be negative
Sum of probabilities < 1
Mean = 0 always
Explanation - Probabilities cannot be negative, so the PDF or PMF must be non-negative for all possible values.
Correct answer is: f(x) ≥ 0 for all x
Q.21 Which of the following distributions is memoryless?
Geometric distribution
Binomial distribution
Normal distribution
Uniform distribution
Explanation - The geometric distribution has the memoryless property: the probability of success in future trials does not depend on past failures.
Correct answer is: Geometric distribution
Q.22 The probability mass function (PMF) is associated with:
Discrete random variables
Continuous random variables
Both
None
Explanation - PMF gives the probability of each possible discrete outcome of a discrete random variable.
Correct answer is: Discrete random variables
Q.23 Which distribution is often used to approximate the binomial distribution for large n and p not near 0 or 1?
Normal distribution
Poisson distribution
Exponential distribution
Uniform distribution
Explanation - By the central limit theorem, the binomial distribution can be approximated by a normal distribution for large n.
Correct answer is: Normal distribution
Q.24 The Poisson distribution can be derived as a limit of which distribution?
Binomial distribution
Normal distribution
Geometric distribution
Uniform distribution
Explanation - Poisson distribution is the limit of the binomial distribution when n is large and p is small such that np = λ.
Correct answer is: Binomial distribution
Q.25 If the probability density function of a continuous random variable X is f(x) = 2x for 0 < x < 1, then the probability P(0.5 < X < 1) is:
0.75
0.25
0.5
1
Explanation - Integrate f(x) from 0.5 to 1: ∫0.5^1 2x dx = [x^2]0.5^1 = 1 - 0.25 = 0.75.
Correct answer is: 0.75