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

What is the probability that a randomly selected value from a standard normal distribution will fall between -1 and 1?

  1. 0.3413

  2. 0.6826

  3. 0.9545

  4. 0.9973
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability that a randomly selected value from a standard normal distribution will fall between -1 and 1 is given by the area under the normal curve between these two values. This area can be calculated using the standard normal distribution table or a calculator and is approximately 0.6826.

Multiple choice

If X is a random variable following a standard normal distribution, what is the probability that X will be less than -2?

  1. 0.0228

  2. 0.05

  3. 0.1587

  4. 0.3413
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the probability that X will be less than -2, we look up the probability that a standard normal random variable will be less than -2 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

Multiple choice

A normal distribution has a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected value from this distribution will be between 80 and 120?

  1. 0.3413

  2. 0.6826

  3. 0.8413

  4. 0.9545
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the probability that a randomly selected value from this distribution will be between 80 and 120, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (80 - 100) / 15 = -1.33 and Z2 = (120 - 100) / 15 = 1.33. Then, we look up the probability that a standard normal random variable will be between -1.33 and 1.33 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.

Multiple choice

If X is a random variable following a standard normal distribution, what is the probability that X will be greater than 1?

  1. 0.1587

  2. 0.3413

  3. 0.5

  4. 0.8413
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the probability that X will be greater than 1, we look up the probability that a standard normal random variable will be greater than 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.1587.

Multiple choice

A normal distribution has a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected value from this distribution will be less than 80?

  1. 0.0228

  2. 0.05

  3. 0.1587

  4. 0.3413
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the probability that a randomly selected value from this distribution will be less than 80, we first standardize the value using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z = (80 - 100) / 15 = -1.33. Then, we look up the probability that a standard normal random variable will be less than -1.33 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

Multiple choice

If X is a random variable following a standard normal distribution, what is the probability that X will be between -1 and 1?

  1. 0.3413

  2. 0.6826

  3. 0.8413

  4. 0.9545
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To find the probability that X will be between -1 and 1, we look up the probability that a standard normal random variable will be between -1 and 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.6826.

Multiple choice

A normal distribution has a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected value from this distribution will be between 70 and 130?

  1. 0.3413

  2. 0.6826

  3. 0.8413

  4. 0.9545
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the probability that a randomly selected value from this distribution will be between 70 and 130, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (70 - 100) / 15 = -2 and Z2 = (130 - 100) / 15 = 2. Then, we look up the probability that a standard normal random variable will be between -2 and 2 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.

Multiple choice

A continuous random variable X has a probability density function given by f(x) = 2x for 0 ≤ x ≤ 1. Find the probability that X is less than or equal to 0.5.

  1. 0.25

  2. 0.5

  3. 0.75

  4. 1
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability that X is less than or equal to 0.5 is given by the integral of f(x) from 0 to 0.5: P(X ≤ 0.5) = ∫[0, 0.5] 2x dx = [x^2]_[0, 0.5] = 0.5^2 - 0^2 = 0.25.

Multiple choice

A company claims that their new product will last for at least 100 hours. A consumer group tests the product and finds that the average lifetime is 95 hours with a standard deviation of 10 hours. Assuming the lifetime follows a normal distribution, what is the probability that a randomly selected product will last for more than 100 hours?

  1. 0.1587

  2. 0.8413

  3. 0.3413

  4. 0.6587
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We can use the standard normal distribution (Z) to calculate this probability: P(X > 100) = P((X - 95)/10 > (100 - 95)/10) = P(Z > 0.5) = 1 - P(Z ≤ 0.5) = 1 - 0.6915 = 0.1587.

Multiple choice

A company claims that their new product will last for at least 100 hours. A consumer group tests the product and finds that the average lifetime is 95 hours with a standard deviation of 10 hours. Assuming the lifetime follows a normal distribution, what is the probability that a randomly selected product will last for less than 80 hours?

  1. 0.0228

  2. 0.1587

  3. 0.3413

  4. 0.8413
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We can use the standard normal distribution (Z) to calculate this probability: P(X < 80) = P((X - 95)/10 < (80 - 95)/10) = P(Z < -1.5) = 0.0228.

Multiple choice

What is the probability of exactly k events occurring in a fixed interval of time or space, given that the average rate of occurrence is lambda?

  1. P(X = k) = (lambda^k * e^(-lambda)) / k!

  2. P(X = k) = (lambda^k * e^(-lambda)) / (k + 1)!

  3. P(X = k) = (lambda^k * e^(-lambda)) / k

  4. P(X = k) = (lambda^k * e^(-lambda)) / (k - 1)!
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of exactly k events occurring in a fixed interval of time or space, given that the average rate of occurrence is lambda, is given by the Poisson distribution formula: P(X = k) = (lambda^k * e^(-lambda)) / k!

Multiple choice

What is the probability of no events occurring in a fixed interval of time or space, given that the average rate of occurrence is lambda?

  1. P(X = 0) = e^(-lambda)

  2. P(X = 0) = 1 - e^(-lambda)

  3. P(X = 0) = lambda * e^(-lambda)

  4. P(X = 0) = 1 - lambda * e^(-lambda)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of no events occurring in a fixed interval of time or space, given that the average rate of occurrence is lambda, is given by P(X = 0) = e^(-lambda).

Multiple choice

What is the probability of at least one event occurring in a fixed interval of time or space, given that the average rate of occurrence is lambda?

  1. P(X >= 1) = 1 - e^(-lambda)

  2. P(X >= 1) = e^(-lambda)

  3. P(X >= 1) = lambda * e^(-lambda)

  4. P(X >= 1) = 1 - lambda * e^(-lambda)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The probability of at least one event occurring in a fixed interval of time or space, given that the average rate of occurrence is lambda, is given by P(X >= 1) = 1 - e^(-lambda).

Multiple choice

A call center receives an average of 10 calls per hour. What is the probability that the call center receives exactly 12 calls in the next hour?

  1. 0.125

  2. 0.25

  3. 0.375

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

Let X be the number of calls received in the next hour. Then X follows a Poisson distribution with a mean of 10. The probability of receiving exactly 12 calls is given by P(X = 12) = (10^12 * e^(-10)) / 12! = 0.125.

Multiple choice

A machine produces defective items with a probability of 0.05. What is the probability that the machine produces exactly 2 defective items in the next 100 items?

  1. 0.102

  2. 0.204

  3. 0.306

  4. 0.408
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let X be the number of defective items produced in the next 100 items. Then X follows a Poisson distribution with a mean of 5 (0.05 * 100). The probability of producing exactly 2 defective items is given by P(X = 2) = (5^2 * e^(-5)) / 2! = 0.204.