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

In a t-distribution with 'ν' degrees of freedom, what is the probability of obtaining a value less than 't_0.05,ν'?

  1. 0.05

  2. 0.95

  3. 1

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a t-distribution with 'ν' degrees of freedom, the probability of obtaining a value less than 't_0.05,ν' is 0.05. This is because the t-distribution is used for hypothesis testing, and the value 't_0.05,ν' is the critical value for a significance level of 0.05.

Multiple choice

In a non-central chi-square distribution with 'k' degrees of freedom and non-centrality parameter 'λ', what is the probability of obtaining a value less than 'χ²_0.05,k,λ'?

  1. 0.05

  2. 0.95

  3. 1

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a non-central chi-square distribution with 'k' degrees of freedom and non-centrality parameter 'λ', the probability of obtaining a value less than 'χ²_0.05,k,λ' is 0.05. This is because the non-central chi-square distribution is used for hypothesis testing, and the value 'χ²_0.05,k,λ' is the critical value for a significance level of 0.05.

Multiple choice

In a non-central t-distribution with 'ν' degrees of freedom and non-centrality parameter 'δ', what is the probability of obtaining a value less than 't_0.05,ν,δ'?

  1. 0.05

  2. 0.95

  3. 1

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a non-central t-distribution with 'ν' degrees of freedom and non-centrality parameter 'δ', the probability of obtaining a value less than 't_0.05,ν,δ' is 0.05. This is because the non-central t-distribution is used for hypothesis testing, and the value 't_0.05,ν,δ' is the critical value for a significance level of 0.05.

Multiple choice

What is the formula for calculating the expected value of a random variable?

  1. E(X) = ∫x f(x) dx

  2. E(X) = ∑x f(x)

  3. E(X) = x̄

  4. E(X) = σ
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expected value of a random variable X is calculated by integrating the product of X and its probability density function (f(x)) over the entire range of possible values of X.

Multiple choice

What is the formula for calculating the probability of an event occurring?

  1. P(A) = 1 - P(¬A)

  2. P(A) = P(B) + P(C)

  3. P(A) = P(B) / P(C)

  4. P(A) = P(B) * P(C)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of an event A occurring is calculated as 1 minus the probability of the event not occurring (¬A).

Multiple choice

In a medical test, the probability of a person having a disease given that the test result is positive is 0.9. The probability of a person having the disease is 0.05. What is the probability of a person having the disease given that the test result is negative?

  1. 0.045

  2. 0.005

  3. 0.95

  4. 0.995
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using Bayes' Theorem, we can calculate the probability of a person having the disease given that the test result is negative as follows: P(Disease|Negative Test) = P(Negative Test|Disease) * P(Disease) / P(Negative Test) = 0.1 * 0.05 / (1 - 0.9) = 0.005.

Multiple choice

In a town, 60% of the population are women and 40% are men. If 70% of the women and 30% of the men own a car, what is the probability that a randomly selected person from the town owns a car?

  1. 0.48

  2. 0.52

  3. 0.62

  4. 0.38
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To calculate the probability, we need to consider the proportion of women and men in the town and their respective probabilities of owning a car. The probability of a randomly selected person owning a car can be calculated as follows: P(Owns Car) = P(Woman and Owns Car) + P(Man and Owns Car) = (0.6 * 0.7) + (0.4 * 0.3) = 0.42 + 0.12 = 0.52.

Multiple choice

A weather forecaster predicts a 30% chance of rain tomorrow. If it does rain, there is a 70% chance of traffic congestion. If it does not rain, there is a 20% chance of traffic congestion. What is the probability of traffic congestion tomorrow?

  1. 0.34

  2. 0.26

  3. 0.44

  4. 0.16
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To calculate the probability of traffic congestion, we need to consider the probability of rain and the respective probabilities of traffic congestion given rain or no rain. The probability of traffic congestion can be calculated as follows: P(Traffic Congestion) = P(Rain and Traffic Congestion) + P(No Rain and Traffic Congestion) = (0.3 * 0.7) + (0.7 * 0.2) = 0.21 + 0.14 = 0.34.

Multiple choice

A company has two production lines, A and B. Line A produces 60% of the company's output, and Line B produces the remaining 40%. Line A has a 2% defect rate, while Line B has a 4% defect rate. If a randomly selected product is defective, what is the probability that it came from Line A?

  1. 0.6

  2. 0.4

  3. 0.72

  4. 0.28
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To calculate the probability that a defective product came from Line A, we need to consider the proportion of output from each line and their respective defect rates. The probability can be calculated as follows: P(Defective from Line A) = P(Line A and Defective) / P(Defective) = (0.6 * 0.02) / (0.6 * 0.02 + 0.4 * 0.04) = 0.012 / 0.016 = 0.72.

Multiple choice

A medical test has a 99% sensitivity and a 95% specificity. If the prevalence of a disease in a population is 1%, what is the probability that a randomly selected person who tests positive actually has the disease?

  1. 0.99

  2. 0.95

  3. 0.94

  4. 0.05
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To calculate the probability that a person who tests positive actually has the disease, we need to consider the sensitivity and specificity of the test, as well as the prevalence of the disease. The probability can be calculated using Bayes' Theorem as follows: P(Disease|Positive Test) = P(Positive Test|Disease) * P(Disease) / P(Positive Test) = 0.99 * 0.01 / (0.99 * 0.01 + 0.05 * 0.99) = 0.94.

Multiple choice

A company has two production lines, A and B. Line A produces 70% of the company's output, and Line B produces the remaining 30%. Line A has a 5% defect rate, while Line B has a 2% defect rate. If a randomly selected product is defective, what is the probability that it came from Line B?

  1. 0.3

  2. 0.7

  3. 0.2

  4. 0.5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To calculate the probability that a defective product came from Line B, we need to consider the proportion of output from each line and their respective defect rates. The probability can be calculated as follows: P(Defective from Line B) = P(Line B and Defective) / P(Defective) = (0.3 * 0.02) / (0.7 * 0.05 + 0.3 * 0.02) = 0.006 / 0.008 = 0.2.

Multiple choice

A medical test has a 98% sensitivity and a 90% specificity. If the prevalence of a disease in a population is 2%, what is the probability that a randomly selected person who tests positive actually has the disease?

  1. 0.98

  2. 0.90

  3. 0.86

  4. 0.14
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To calculate the probability that a person who tests positive actually has the disease, we need to consider the sensitivity and specificity of the test, as well as the prevalence of the disease. The probability can be calculated using Bayes' Theorem as follows: P(Disease|Positive Test) = P(Positive Test|Disease) * P(Disease) / P(Positive Test) = 0.98 * 0.02 / (0.98 * 0.02 + 0.1 * 0.98) = 0.86.

Multiple choice

In a queueing system, what is the probability that a customer will have to wait for service?

  1. P(W > 0)

  2. P(W < 0)

  3. P(W = 0)

  4. P(W = \infty)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that a customer will have to wait for service is denoted by P(W > 0), where W is the waiting time.

Multiple choice

In a queueing system, what is the probability that the system is empty?

  1. P(N = 0)

  2. P(N > 0)

  3. P(N < 0)

  4. P(N = \infty)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that the system is empty is denoted by P(N = 0), where N is the number of customers in the system.

Multiple choice

What is a probability space in Probability Theory?

  1. A set of all possible outcomes of an experiment.

  2. A set of all possible events in an experiment.

  3. A triple consisting of a sample space, a sigma-algebra of events, and a probability measure.

  4. A function that assigns a probability to each event in an experiment.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A probability space is a mathematical model that describes the behavior of a random experiment. It consists of a sample space, which is the set of all possible outcomes of the experiment, a sigma-algebra of events, which is a collection of subsets of the sample space, and a probability measure, which assigns a probability to each event in the sigma-algebra.