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 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

In the Atkinson model, the probability of success is determined by:

  1. The strength of the motive

  2. The difficulty of the task

  3. Both the strength of the motive and the difficulty of the task

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

In the Atkinson model, the probability of success is determined by both the strength of the motive and the difficulty of the task.

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

Which of the following is NOT a common type of mechanical betting system based on probability distributions?

  1. Poisson Distribution System

  2. Normal Distribution System

  3. Binomial Distribution System

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

The Martingale System is a negative progression betting system based on the principle of doubling the bet size after each loss. It is not a common type of mechanical betting system based on probability distributions, as it does not rely on statistical analysis to make betting decisions.

Multiple choice

What is the typical acceptance probability of a worse solution in Simulated Annealing?

  1. It is always accepted.

  2. It is always rejected.

  3. It depends on the temperature parameter.

  4. It depends on the difference between the current solution and the worse solution.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The acceptance probability of a worse solution in Simulated Annealing depends on the temperature parameter. At higher temperatures, worse solutions are more likely to be accepted, while at lower temperatures, worse solutions are less likely to be accepted.

Multiple choice

What are the chances of getting a new trial?

  1. Less than 1%

  2. 1-5%

  3. 5-10%

  4. 10-20%
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chances of getting a new trial are very low. According to the National Registry of Exonerations, only about 1-5% of defendants who file a motion for a new trial are successful.

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.

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 greater than 10?

  1. 0.1587

  2. 0.3173

  3. 0.4772

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

The probability that $X$ takes on a value greater than 10 is given by the area under the normal distribution curve to the right of 10. This area is approximately 0.3173.