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

The probability of an event occurring is always between 0 and 1. True or False?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of an event occurring can never be less than 0 or greater than 1.

Multiple choice

According to the propensity interpretation of probability, what is the probability of an event?

  1. The frequency of the event in a long series of trials.

  2. The degree of belief in the occurrence of the event.

  3. The disposition of a system to produce the event under specified conditions.

  4. The logical consequence of the evidence supporting the event.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The propensity interpretation of probability views probability as a measure of the disposition of a system to produce a particular event under specified conditions. It is based on the idea that probabilities are inherent properties of systems and not merely reflections of our beliefs or knowledge.

Multiple choice

Which of the following is not a valid probability distribution?

  1. Uniform distribution

  2. Normal distribution

  3. Binomial distribution

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

The Cauchy distribution is not a valid probability distribution because its cumulative distribution function is not defined at all points. This means that it is not possible to assign probabilities to all possible outcomes in the sample space.

Multiple choice

The Expected Value of a random variable is:

  1. The sum of all possible outcomes multiplied by their probabilities

  2. The difference between the maximum and minimum possible outcomes

  3. The probability of the most likely outcome

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Expected Value of a random variable is the weighted average of all possible outcomes, where the weights are the probabilities of each outcome.

Multiple choice

What is the probability of obtaining a defective item from a production process if the probability of a non-defective item is 0.9?

  1. 0.1

  2. 0.2

  3. 0.3

  4. 0.4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of obtaining a defective item is the complement of the probability of obtaining a non-defective item. Therefore, the probability of a defective item is 1 - 0.9 = 0.1.

Multiple choice

What is the name of the book that R. Parthasarathy co-authored with K.R. Parthasarathy in 1992, which provides an introduction to probability theory?

  1. Probability Theory: An Introduction

  2. Stochastic Processes: An Introduction

  3. Statistical Mechanics: An Introduction

  4. Quantum Mechanics: An Introduction
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The book, Probability Theory: An Introduction, by R. Parthasarathy and K.R. Parthasarathy, was published in 1992.

Multiple choice

According to signal detection theory, what is the relationship between the signal-to-noise ratio and the probability of detection?

  1. The higher the signal-to-noise ratio, the higher the probability of detection

  2. The lower the signal-to-noise ratio, the higher the probability of detection

  3. The signal-to-noise ratio has no effect on the probability of detection
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to signal detection theory, the higher the signal-to-noise ratio, the more likely it is that the signal will be detected.

Multiple choice

In signal detection theory, the probability of a hit is defined as the probability of:

  1. Correctly identifying a signal when it is present.

  2. Incorrectly identifying a signal when it is absent.

  3. Correctly identifying a noise when it is present.

  4. Incorrectly identifying a noise when it is absent.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In signal detection theory, the probability of a hit is the probability of correctly identifying a signal when it is present.

Multiple choice

In signal detection theory, the probability of a false alarm is defined as the probability of:

  1. Correctly identifying a signal when it is present.

  2. Incorrectly identifying a signal when it is absent.

  3. Correctly identifying a noise when it is present.

  4. Incorrectly identifying a noise when it is absent.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In signal detection theory, the probability of a false alarm is the probability of incorrectly identifying a signal when it is absent.

Multiple choice

A farmer is considering two different crop varieties, A and B. Variety A has a yield distribution with a mean of 100 bushels per acre and a standard deviation of 20 bushels per acre. Variety B has a yield distribution with a mean of 120 bushels per acre and a standard deviation of 30 bushels per acre. Which variety has the higher probability of yielding at least 125 bushels per acre?

  1. Variety A

  2. Variety B

  3. Both varieties have the same probability

  4. Cannot be determined from the given information
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of obtaining a yield of at least 125 bushels per acre can be calculated using the standard normal distribution. The z-score for a yield of 125 bushels per acre for Variety A is (125 - 100) / 20 = 1.25. The z-score for a yield of 125 bushels per acre for Variety B is (125 - 120) / 30 = 0.17. The probability of obtaining a z-score of 0.17 or higher is 0.5675. The probability of obtaining a z-score of 1.25 or higher is 0.1056. Therefore, Variety B has the higher probability of yielding at least 125 bushels per acre.

Multiple choice

If the expected value of a random variable $X$ is 0 and its variance is 1, what is the probability that $X$ takes on a value between -1 and 1?

  1. 0.6827

  2. 0.9545

  3. 0.9973

  4. 0.7357
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that $X$ takes on a value between -1 and 1 is given by the area under the normal distribution curve between -1 and 1. This area is approximately 0.6827.

Multiple choice

If the expected value of a random variable $X$ is 10 and its variance is 4, what is the probability that $X$ takes on a value greater than 12?

  1. 0.1587

  2. 0.3173

  3. 0.4772

  4. 0.6331
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that $X$ takes on a value greater than 12 is given by the area under the normal distribution curve to the right of 12. This area is approximately 0.1587.

Multiple choice

If the expected value of a random variable $X$ is 5 and its variance is 9, what is the probability that $X$ takes on a value between 2 and 8?

  1. 0.6827

  2. 0.9545

  3. 0.9973

  4. 0.7357
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability that $X$ takes on a value between 2 and 8 is given by the area under the normal distribution curve between 2 and 8. This area is approximately 0.9545.

Multiple choice

If the expected value of a random variable $X$ is 0 and its variance is 1, what is the probability that $X$ takes on a value less than -2?

  1. 0.0228

  2. 0.0455

  3. 0.0668

  4. 0.0893
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that $X$ takes on a value less than -2 is given by the area under the normal distribution curve to the left of -2. This area is approximately 0.0228.

Multiple choice

If the expected value of a random variable $X$ is 10 and its variance is 4, what is the probability that $X$ takes on a value between 8 and 12?

  1. 0.6827

  2. 0.9545

  3. 0.9973

  4. 0.7357
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability that $X$ takes on a value between 8 and 12 is given by the area under the normal distribution curve between 8 and 12. This area is approximately 0.6827.