Mathematics

Probability Distributions

457 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 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).

Multiple choice

What is the expected value of a random variable X with probability density function f(x) = \frac{1}{2}e^{-|x|}?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The expected value of a random variable X with probability density function f(x) is defined as E(X) = ∫xf(x)dx. In this case, E(X) = ∫x\frac{1}{2}e^{-|x|}dx = 1.

Multiple choice

Given two independent events A and B with P(A) = 0.4 and P(B) = 0.6, what is the probability of both A and B occurring?

  1. 0.2

  2. 0.24

  3. 0.36

  4. 0.48
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of both A and B occurring is given by P(A ∩ B) = P(A)P(B). Therefore, P(A ∩ B) = 0.4 * 0.6 = 0.24.

Multiple choice

Which of the following is an example of a discrete random variable?

  1. Height of a randomly selected person

  2. Number of heads in 10 coin tosses

  3. Stock prices over time

  4. Number of customers in a store at a given time
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A discrete random variable can take on only a countable number of values. The number of heads in 10 coin tosses is an example of a discrete random variable because it can take on only the values 0, 1, 2, ..., 10.

Multiple choice

What is the expected value of a random variable X with probability mass function P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, and 6?

  1. 3.5

  2. 4

  3. 4.5

  4. 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(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5.

Multiple choice

A survey of 100 people found that 60 people like chocolate, 40 people like vanilla, and 20 people like both chocolate and vanilla. What is the probability that a randomly selected person from the survey likes chocolate or vanilla?

  1. 0.8

  2. 0.6

  3. 0.4

  4. 0.2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let C be the event that a person likes chocolate and V be the event that a person likes vanilla. Then, P(C or V) = P(C) + P(V) - P(C and V) = 60/100 + 40/100 - 20/100 = 80/100 = 0.8.

Multiple choice

A company has a 10% chance of winning a contract. If the company submits 10 bids, what is the probability that they will win at least one contract?

  1. 0.63

  2. 0.74

  3. 0.85

  4. 0.95
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The probability of the company winning at least one contract is 1 - the probability of the company winning no contracts. The probability of the company winning no contracts is (0.9)^10 = 0.3487. Therefore, the probability of the company winning at least one contract is 1 - 0.3487 = 0.95.

Multiple choice

A machine produces defective items with a probability of 0.05. If 100 items are produced, what is the probability that exactly 5 items are defective?

  1. 0.074

  2. 0.135

  3. 0.201

  4. 0.268
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

This is a binomial distribution problem. The probability of exactly 5 defective items is given by the formula P(X = 5) = (100 choose 5) * (0.05)^5 * (0.95)^95 = 0.201.

Multiple choice

A box contains 100 light bulbs, of which 10 are defective. If 10 light bulbs are randomly selected from the box, what is the probability that exactly 2 of them are defective?

  1. 0.252

  2. 0.308

  3. 0.364

  4. 0.420
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is a hypergeometric distribution problem. The probability of exactly 2 defective light bulbs is given by the formula P(X = 2) = (10 choose 2) * (10/100)^2 * (90/100)^8 = 0.308.