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

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 -3 and 3?

  1. 0.6827

  2. 0.9545

  3. 0.9973

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

The probability that $X$ takes on a value between -3 and 3 is given by the area under the normal distribution curve between -3 and 3. This area is approximately 0.9973.

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 less than 8?

  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 less than 8 is given by the area under the normal distribution curve to the left of 8. 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 0 and 10?

  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 0 and 10 is given by the area under the normal distribution curve between 0 and 10. 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 greater than 2?

  1. 0.0228

  2. 0.0455

  3. 0.0668

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

The probability that $X$ takes on a value greater than 2 is given by the area under the normal distribution curve to the right of 2. This area is approximately 0.0455.

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 6 and 14?

  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 6 and 14 is given by the area under the normal distribution curve between 6 and 14. This area is approximately 0.9545.

Multiple choice

What is the statement of Chebyshev's inequality?

  1. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \ge k\sigma\) is at most \(1/k^2\).
  2. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \le k\sigma\) is at most \(1/k^2\).
  3. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \ge k\sigma\) is at least \(1/k^2\).
  4. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \le k\sigma\) is at least \(1/k^2\).
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Chebyshev's inequality states that for any random variable (X) with mean (\mu) and variance (\sigma^2), the probability that the absolute value of the difference between (X) and (\mu) is greater than or equal to (k\sigma) is at most (1/k^2).

Multiple choice

What is the statement of Markov's inequality?

  1. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \ge k\sigma\) is at most \(1/k^2\).
  2. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \le k\sigma\) is at most \(1/k^2\).
  3. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \ge k\sigma\) is at least \(1/k^2\).
  4. For any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(|X - \mu| \le k\sigma\) is at least \(1/k^2\).
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Markov's inequality states that for any random variable (X) with mean (\mu) and variance (\sigma^2), the probability that the absolute value of the difference between (X) and (\mu) is greater than or equal to (k\sigma) is at least (1/k^2).

Multiple choice

What is the probability of an event that is certain to occur?

  1. 0

  2. 1

  3. 0.5

  4. It depends on the context
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of an event that is certain to occur is 1, since it will definitely happen.

Multiple choice

What is the probability of an event that is impossible to occur?

  1. 0

  2. 1

  3. 0.5

  4. It depends on the context
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of an event that is impossible to occur is 0, since it will never happen.

Multiple choice

What is the probability of an event that is equally likely to occur or not occur?

  1. 0

  2. 1

  3. 0.5

  4. It depends on the context
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The probability of an event that is equally likely to occur or not occur is 0.5, since there are two equally likely outcomes.

Multiple choice

What is the probability of an event A given that event B has already occurred?

  1. P(A)

  2. P(B)

  3. P(A|B)

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

The probability of an event A given that event B has already occurred is denoted as P(A|B) and is known as conditional probability.

Multiple choice

What is a probability distribution?

  1. A function that assigns probabilities to events

  2. A function that assigns probabilities to random variables

  3. A function that assigns probabilities to both events and random variables

  4. A function that assigns probabilities to none of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A probability distribution is a function that assigns probabilities to random variables, specifying the likelihood of different outcomes.

Multiple choice

What is the binomial distribution?

  1. A discrete probability distribution that describes the number of successes in a sequence of independent experiments

  2. A continuous probability distribution that describes the number of successes in a sequence of independent experiments

  3. A discrete probability distribution that describes the time until the first success in a sequence of independent experiments

  4. A continuous probability distribution that describes the time until the first success in a sequence of independent experiments
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent experiments, each of which has a constant probability of success.

Multiple choice

What is the Poisson distribution?

  1. A discrete probability distribution that describes the number of events that occur in a fixed interval of time or space

  2. A continuous probability distribution that describes the number of events that occur in a fixed interval of time or space

  3. A discrete probability distribution that describes the time until the first event occurs in a fixed interval of time or space

  4. A continuous probability distribution that describes the time until the first event occurs in a fixed interval of time or space
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Poisson distribution is a discrete probability distribution that describes the number of events that occur in a fixed interval of time or space, where the events occur independently and at a constant rate.

Multiple choice

What is the uniform distribution?

  1. A continuous probability distribution that is constant over a specified interval

  2. A discrete probability distribution that is constant over a specified interval

  3. A continuous probability distribution that is skewed to the right

  4. A discrete probability distribution that is skewed to the right
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The uniform distribution is a continuous probability distribution that is constant over a specified interval, indicating that all values within the interval are equally likely to occur.