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

Which of the following properties is not satisfied by a Lévy process?

  1. Independent increments

  2. Stationary increments

  3. Gaussian distribution

  4. Infinitely divisible distribution
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Lévy processes do not necessarily have a Gaussian distribution. In fact, many Lévy processes, such as the Poisson process, have distributions that are far from Gaussian.

Multiple choice

What is a probability distribution?

  1. A function that describes the probability of a random variable taking on a particular value.

  2. A function that describes the probability of a random variable taking on a range of values.

  3. A function that describes the probability of a random variable taking on a set of values.

  4. A function that describes the probability of a random variable taking on a sequence of values.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A probability distribution is a function that describes the probability of a random variable taking on a particular value. Probability distributions are used in statistics, probability theory, and other applications.

Multiple choice

What is the probability density function of a continuous random variable?

  1. A function that gives the probability of a random variable taking on a specific value.

  2. A function that gives the probability of a random variable taking on a range of values.

  3. A function that gives the probability of a random variable taking on a set of values.

  4. A function that gives the probability of a random variable taking on an interval of values.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability density function of a continuous random variable is a function that gives the probability of a random variable taking on a range of values.

Multiple choice

What is the cumulative distribution function of a random variable?

  1. A function that gives the probability of a random variable taking on a specific value or less.

  2. A function that gives the probability of a random variable taking on a range of values or less.

  3. A function that gives the probability of a random variable taking on a set of values or less.

  4. A function that gives the probability of a random variable taking on an interval of values or less.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The cumulative distribution function of a random variable is a function that gives the probability of a random variable taking on a specific value or less.

Multiple choice

Which of the following is not a type of probability distribution?

  1. Normal distribution

  2. Binomial distribution

  3. Poisson distribution

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

Uniform distribution is not a probability distribution, but rather a type of continuous distribution.

Multiple choice

Which of the following is not a type of probability distribution?

  1. Normal distribution

  2. Binomial distribution

  3. Poisson distribution

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

Uniform distribution is not a probability distribution, but rather a type of continuous distribution.

Multiple choice

In a Markov chain, what is the probability of transitioning from state $i$ to state $j$ in $n$ steps?

  1. $P_{ij}^n$
  2. $P_{ij}^{n-1}$
  3. $P_{ij}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a Markov chain, the probability of transitioning from state $i$ to state $j$ in $n$ steps is given by the $n$-th power of the transition probability matrix $P$, denoted as $P_{ij}^n$.

Multiple choice

In a Markov chain model of behavior change, what is the probability of being in state $j$ at time $t$ given that the system started in state $i$ at time $0$?

  1. $P_{ij}(t)$
  2. $P_{ji}(t)$
  3. $P_{ii}(t)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a Markov chain model of behavior change, the probability of being in state $j$ at time $t$ given that the system started in state $i$ at time $0$ is given by $P_{ij}(t)$, which is an element of the transition probability matrix raised to the power of $t$.

Multiple choice

In probability, what is the probability of an event occurring?

  1. The ratio of the number of favorable outcomes to the total number of possible outcomes

  2. The ratio of the number of unfavorable outcomes to the total number of possible outcomes

  3. The difference between the number of favorable outcomes and the number of unfavorable outcomes

  4. The sum of the number of favorable outcomes and the number of unfavorable outcomes
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of an event occurring is the ratio of the number of favorable outcomes to the total number of possible outcomes.

Multiple choice

In probability theory, what does the symbol '0' often represent?

  1. An impossible event

  2. A certain event

  3. A random event

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

In probability, the symbol '0' is commonly used to represent an impossible event. An impossible event is an event that cannot occur under any circumstances.

Multiple choice

What is the probability of an impossible event?

  1. 1

  2. 0

  3. 0.5

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

The probability of an impossible event is always 0. This is because an impossible event cannot occur, so the probability of its occurrence is zero.

Multiple choice

What is the probability of a certain event?

  1. 1

  2. 0

  3. 0.5

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

The probability of a certain event is always 1. This is because a certain event is guaranteed to occur, so the probability of its occurrence is 1.

Multiple choice

In a probability distribution, what does the value of '0' often indicate?

  1. The mean of the distribution

  2. The median of the distribution

  3. The mode of the distribution

  4. The probability of an impossible event
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a probability distribution, the value of '0' often indicates the probability of an impossible event. This is because an impossible event has a probability of zero.

Multiple choice

What is the expected value of a random variable that takes the value '0' with probability 'p'?

  1. 0

  2. p

  3. 1-p

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

The expected value of a random variable that takes the value '0' with probability 'p' is 0. This is because the expected value is calculated by multiplying each possible value of the random variable by its probability and then summing the results. In this case, the only possible value is '0', and its probability is 'p', so the expected value is 0 * p = 0.

Multiple choice

In a binomial distribution with parameters 'n' and 'p', what is the probability of obtaining exactly '0' successes?

  1. (1-p)^n

  2. p^n

  3. n * p

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

In a binomial distribution with parameters 'n' and 'p', the probability of obtaining exactly '0' successes is given by (1-p)^n. This is because the probability of success is 'p' and the probability of failure is 1-p. To obtain exactly '0' successes, we need to have 'n' consecutive failures, which has a probability of (1-p)^n.