Mathematics

Probability Distributions

488 Questions

Probability distributions describe the likelihood of different outcomes in a statistical experiment. Key topics include calculating confidence intervals, understanding Type II errors, and working with the standard normal distribution. These concepts are frequently tested in mathematics and statistics sections of various competitive exams.

Standard normal distributionType II error probabilityConfidence interval interpretationPoisson distributionProbability density function

Probability Distributions Questions

Multiple choice

What is the probability of getting a z-score of 1.96 or higher in a standard normal distribution?

  1. 0.025

  2. 0.05

  3. 0.01

  4. 0.005
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The z-score of 1.96 corresponds to the 97.5th percentile in a standard normal distribution. So, the probability of getting a z-score of 1.96 or higher is 0.025.

Multiple choice

What is the probability of getting a sample proportion of 0.5 or higher from a population with a proportion of 0.5, if the sample size is 100?

  1. 0.5

  2. 0.68

  3. 0.95

  4. 0.99
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The sampling distribution of the sample proportion is a normal distribution with a mean equal to the population proportion and a standard deviation equal to the square root of the population proportion times the square root of 1 minus the population proportion, divided by the square root of the sample size. In this case, the sampling distribution of the sample proportion has a mean of 0.5 and a standard deviation of sqrt(0.5 * 0.5) / sqrt(100) = 0.05. So, the probability of getting a sample proportion of 0.5 or higher is 0.99.

Multiple choice

What is the probability of getting a chi-square statistic of 10 or higher from a chi-square distribution with 5 degrees of freedom?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chi-square distribution with 5 degrees of freedom has a mean of 5 and a standard deviation of sqrt(2 * 5) = sqrt(10). So, the probability of getting a chi-square statistic of 10 or higher is 0.05.

Multiple choice

What is the probability of getting a t-statistic of 2 or higher from a t-distribution with 10 degrees of freedom?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The t-distribution with 10 degrees of freedom has a mean of 0 and a standard deviation of sqrt(10 / (10 - 2)) = sqrt(10 / 8) = sqrt(5 / 4). So, the probability of getting a t-statistic of 2 or higher is 0.05.

Multiple choice

What is the probability of getting an F-statistic of 4 or higher from an F-distribution with 5 and 10 degrees of freedom?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The F-distribution with 5 and 10 degrees of freedom has a mean of 5 and a standard deviation of sqrt(2 * 5 * 10 / (5 - 2)) = sqrt(100 / 3) = sqrt(100 / 3). So, the probability of getting an F-statistic of 4 or higher is 0.05.

Multiple choice

What is the probability of getting a correlation coefficient of 0.5 or higher from a population with a correlation coefficient of 0.5?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sampling distribution of the correlation coefficient is a t-distribution with n-2 degrees of freedom, where n is the sample size. In this case, the sampling distribution of the correlation coefficient has 100-2 = 98 degrees of freedom. The probability of getting a correlation coefficient of 0.5 or higher is 0.10.

Multiple choice

What is the probability of getting a regression coefficient of 2 or higher from a population with a regression coefficient of 2?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sampling distribution of the regression coefficient is a t-distribution with n-2 degrees of freedom, where n is the sample size. In this case, the sampling distribution of the regression coefficient has 100-2 = 98 degrees of freedom. The probability of getting a regression coefficient of 2 or higher is 0.05.

Multiple choice

What is the probability of getting an ANOVA F-statistic of 4 or higher from a population with an ANOVA F-statistic of 4?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sampling distribution of the ANOVA F-statistic is an F-distribution with k-1 and n-k degrees of freedom, where k is the number of groups and n is the total sample size. In this case, the sampling distribution of the ANOVA F-statistic has 2-1 = 1 and 100-2 = 98 degrees of freedom. The probability of getting an ANOVA F-statistic of 4 or higher is 0.05.

Multiple choice

What is the probability of getting a chi-square goodness-of-fit statistic of 10 or higher from a population with a chi-square goodness-of-fit statistic of 10?

  1. 0.05

  2. 0.10

  3. 0.15

  4. 0.20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sampling distribution of the chi-square goodness-of-fit statistic is a chi-square distribution with k-1 degrees of freedom, where k is the number of categories. In this case, the sampling distribution of the chi-square goodness-of-fit statistic has 5-1 = 4 degrees of freedom. The probability of getting a chi-square goodness-of-fit statistic of 10 or higher is 0.05.

Multiple choice

What is the formula for calculating the Population Attributable Risk?

  1. (a / (a + b)) - (c / (c + d))

  2. ((a + b) / (c + d)) - 1

  3. (a - b) / (c - d)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The formula for calculating the Population Attributable Risk is ((a + b) / (c + d)) - 1, where 'a' represents the number of individuals with both exposure and outcome, 'b' represents the number of individuals with exposure but without outcome, 'c' represents the number of individuals without exposure but with outcome, and 'd' represents the number of individuals without exposure and without outcome.

Multiple choice

In a queuing system, what is the probability that a customer will be served within a certain amount of time?

  1. 1 - (Arrival rate / Service rate)

  2. 1 - (Service rate / Arrival rate)

  3. 1 - (Arrival rate / (Arrival rate + Service rate))

  4. 1 - (Service rate / (Service rate + Arrival rate))
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

This is known as the probability of being served within a certain time and is calculated as 1 minus the ratio of the arrival rate to the sum of the arrival rate and the service rate.

Multiple choice

What is the expected value of a random variable X with probability mass function P(X = x) = 1/3 for x = 1, 2, and 3?

  1. 2

  2. 2.5

  3. 3

  4. 3.5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expected value of a random variable X is defined as E(X) = ΣxP(X = x). In this case, E(X) = 1(1/3) + 2(1/3) + 3(1/3) = 2.

Multiple choice

Given two events A and B with P(A) = 0.4, P(B) = 0.6, and P(A ∩ B) = 0.2, what is the probability of A given B?

  1. 0.2

  2. 0.33

  3. 0.5

  4. 0.67
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of A given B is P(A | B) = P(A ∩ B) / P(B). Therefore, P(A | B) = 0.2 / 0.6 = 0.33.

Multiple choice

What is the expected number of arrivals in a Poisson process with rate λ over a time interval of length t?

  1. λt

  2. λt^2

  3. λ^2t

  4. λ^2t^2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expected number of arrivals in a Poisson process with rate λ over a time interval of length t is given by λt.

Multiple choice

In a Markov chain with states S1, S2, and S3, what is the probability of transitioning from state S1 to state S3 in two steps?

  1. P(S1, S2)P(S2, S3)

  2. P(S1, S3)^2

  3. P(S1, S2) + P(S2, S3)

  4. P(S1, S3) - P(S1, S2)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of transitioning from state S1 to state S3 in two steps is given by the product of the transition probabilities P(S1, S2) and P(S2, S3).