Q.1 Which of the following is a valid probability distribution?
Probabilities sum to 0.5
Probabilities sum to 1
Probabilities sum to 2
Probabilities are negative
Explanation - A probability distribution must have probabilities between 0 and 1, and the sum of all probabilities must equal 1.
Correct answer is: Probabilities sum to 1
Q.2 In a discrete probability distribution, the random variable takes values that are:
Continuous
Only integers
Countable
Uncountable
Explanation - Discrete probability distributions are defined over countable values, such as integers or finite sets.
Correct answer is: Countable
Q.3 The mean of a probability distribution is also called:
Median
Expectation
Variance
Mode
Explanation - The expected value or expectation represents the mean of the probability distribution.
Correct answer is: Expectation
Q.4 Which distribution is used to model the number of successes in a fixed number of independent Bernoulli trials?
Normal
Poisson
Binomial
Exponential
Explanation - The binomial distribution models the probability of obtaining a certain number of successes in fixed trials.
Correct answer is: Binomial
Q.5 Which of the following is a continuous probability distribution?
Binomial
Poisson
Normal
Geometric
Explanation - The normal distribution is continuous, whereas binomial, Poisson, and geometric are discrete.
Correct answer is: Normal
Q.6 The variance of a probability distribution measures:
Central tendency
Spread of data
Number of outcomes
Skewness
Explanation - Variance quantifies the dispersion of data points around the mean.
Correct answer is: Spread of data
Q.7 If a fair die is rolled, what is the expected value of the outcome?
2
3
3.5
4
Explanation - The mean of outcomes (1+2+3+4+5+6)/6 = 3.5.
Correct answer is: 3.5
Q.8 Which of these distributions has equal mean, median, and mode?
Normal
Binomial
Poisson
Geometric
Explanation - The normal distribution is symmetric and has mean = median = mode.
Correct answer is: Normal
Q.9 The Poisson distribution is used to model:
Time between events
Number of rare events in fixed interval
Successes in fixed trials
Continuous outcomes
Explanation - Poisson distribution is ideal for modeling the number of rare events in a fixed period of time or space.
Correct answer is: Number of rare events in fixed interval
Q.10 The exponential distribution is related to:
Time until an event occurs
Total number of events
Symmetry of outcomes
Central limit theorem
Explanation - Exponential distribution models the waiting time until the next event in a Poisson process.
Correct answer is: Time until an event occurs
Q.11 In a binomial distribution with parameters n and p, the variance is:
np
np(1-p)
n(1-p)
p(1-p)
Explanation - Variance of a binomial distribution is np(1-p).
Correct answer is: np(1-p)
Q.12 Which probability distribution is often used to approximate the binomial distribution when n is large and p is small?
Normal
Poisson
Exponential
Geometric
Explanation - The Poisson distribution approximates the binomial when n is large and p is small.
Correct answer is: Poisson
Q.13 The normal distribution is also known as:
Bell curve
Step curve
Exponential curve
Geometric curve
Explanation - The normal distribution has a bell-shaped curve symmetric about the mean.
Correct answer is: Bell curve
Q.14 What is the mean of a Poisson distribution with parameter λ?
λ
1/λ
λ^2
√λ
Explanation - The Poisson distribution has both mean and variance equal to λ.
Correct answer is: λ
Q.15 The cumulative distribution function (CDF) gives:
Probability density
Mean of distribution
Probability X ≤ x
Variance
Explanation - The CDF gives the probability that a random variable takes a value less than or equal to x.
Correct answer is: Probability X ≤ x
Q.16 Which distribution models the number of trials needed for the first success?
Binomial
Geometric
Poisson
Exponential
Explanation - The geometric distribution measures how many Bernoulli trials are required before the first success.
Correct answer is: Geometric
Q.17 If X is a standard normal random variable, then E[X] equals:
0
1
-1
0.5
Explanation - The standard normal distribution has mean 0 and variance 1.
Correct answer is: 0
Q.18 For a uniform distribution U(a, b), the mean is:
a+b
(a+b)/2
(a-b)/2
ab
Explanation - The mean of uniform distribution is midpoint of interval = (a+b)/2.
Correct answer is: (a+b)/2
Q.19 The binomial distribution becomes symmetric when:
p=0
p=0.25
p=0.5
p=1
Explanation - When probability of success = 0.5, the distribution is symmetric.
Correct answer is: p=0.5
Q.20 Which distribution is commonly used in reliability analysis?
Poisson
Geometric
Exponential
Normal
Explanation - The exponential distribution is used in reliability and survival analysis to model lifetimes.
Correct answer is: Exponential
Q.21 The normal distribution is fully defined by:
Mean only
Variance only
Mean and variance
Median and mode
Explanation - Normal distribution is characterized by two parameters: mean (μ) and variance (σ²).
Correct answer is: Mean and variance
Q.22 The Poisson distribution is a limit of which distribution?
Uniform
Normal
Binomial
Exponential
Explanation - Poisson is the limiting case of binomial distribution for large n and small p with np = λ.
Correct answer is: Binomial
Q.23 In a standard normal distribution, approximately what percentage of data lies within 1 standard deviation from the mean?
50%
68%
95%
99.7%
Explanation - The empirical rule states about 68% of data lies within ±1σ in a normal distribution.
Correct answer is: 68%
Q.24 Which of the following is NOT a discrete distribution?
Binomial
Poisson
Geometric
Exponential
Explanation - Exponential is continuous; others are discrete distributions.
Correct answer is: Exponential
Q.25 The variance of a uniform distribution U(a, b) is:
(b-a)/2
(b-a)^2/2
(b-a)^2/12
(a+b)^2/12
Explanation - Variance of uniform distribution = (b-a)² / 12.
Correct answer is: (b-a)^2/12