Continuous Probability Distributions: Master the Fundamentals with Interactive Quizzes

Question 1

Which of the following is the probability density function of a normal distribution with mean μ and variance σ²?

Accepted Answer

((2πσ²)⁻¹/²) * exp(-(x - μ)² / (2σ²))

Answer

((2σ²)¹/²) * exp(-(x - μ)² / (2σ²))

Answer

((2πσ)⁻¹/²) * exp(-(x - μ)² / (2σ))

Answer

((2πσ)¹/²) * exp(-(x - μ)² / (2σ))

Question 2

What is the cumulative distribution function of an exponential distribution with parameter λ?

Accepted Answer

1 - exp(-λx)

Answer

λ * exp(-λx)

Answer

exp(-λx)

Answer

1 - λ * exp(-λx)

Question 3

Which of the following is the expected value of a uniform distribution with lower bound a and upper bound b?

Accepted Answer

(a + b) / 2

Answer

a + b

Answer

a - b

Answer

(a - b) / 2

Question 4

Which of the following is the variance of a normal distribution with mean μ and variance σ²?

Accepted Answer

σ²

Answer

μ²

Answer

σ

Answer

μ

Question 5

Which of the following is the median of a uniform distribution between 0 and 1?

Accepted Answer

0.5

Answer

0

Answer

0.25

Answer

1

Question 6

Which of the following is the probability density function of the standard normal distribution?

Accepted Answer

f(x) = (1/√(2π)) * e^(-x^2/2)

Answer

f(x) = (1/σ√(2π)) * e^(-(x-μ)^2/(2σ^2))

Answer

f(x) = (1/b-a) for a < x < b

Answer

f(x) = λ * e^(-λx)

Question 7

What is the mean of the exponential distribution with parameter λ?

Accepted Answer

μ = 1/λ

Answer

μ = λ

Answer

μ = 1/λ^2

Answer

μ = λ^2

Question 8

A random variable X has a uniform distribution between a and b. What is the probability that X takes a value less than (a+b)/2?

Accepted Answer

P(X < (a+b)/2) = 1/2

Answer

P(X < (a+b)/2) = (a+b)/2

Answer

P(X < (a+b)/2) = 1

Answer

P(X < (a+b)/2) = 0

Question 9

Which of the following is a characteristic of the normal distribution?

Accepted Answer

It is approximately bell-shaped.

Answer

It is skewed left.

Answer

It is bimodal.

Answer

It has a finite range.

Question 10

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

Accepted Answer

P(−1 < z < 1) = 0.6827

Answer

P(−1 < z < 1) = 0.0455

Answer

P(−1 < z < 1) = 0.3173

Answer

P(−1 < z < 1) = 0.9545

Question 11

Which of the following is a measure of dispersion for a continuous probability distribution?

Accepted Answer

Variance

Answer

Skewness

Answer

Median

Answer

Mean

Question 12

Which of the following is the cumulative distribution function of the standard uniform distribution?

Accepted Answer

F(x) = x for 0 ≤ x ≤ 1

Answer

F(x) = λ * e^(-λx)

Answer

F(x) = (1/√(2π)) * e^(-x^2/2)

Question 13

What is the relationship between the probability density function and the cumulative distribution function?

Accepted Answer

The cumulative distribution function is the integral of the probability density function.

Answer

The probability density function and the cumulative distribution function are independent.

Answer

The probability density function is the derivative of the cumulative distribution function.

Question 14

Which of the following distributions is a special case of the gamma distribution?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Normal distribution

Answer

Uniform distribution

Question 15

Which of the following is a characteristic of a continuous probability distribution?

Accepted Answer

Can take any value within a specified range

Answer

Only takes discrete values

Answer

Has a finite number of possible values

Answer

Can be represented by a probability mass function

Question 16

Which of the following is a property of the cumulative distribution function (CDF) of a continuous probability distribution?

Accepted Answer

Is a continuous, non-decreasing function

Answer

Is constant over the entire range

Answer

Is a linear function

Answer

Takes negative values

Question 17

The probability density function of a random variable X is given by f(x) = 1/(b-a) for a ≤ x ≤ b. What is the mean of X?

Accepted Answer

(a+b)/2

Answer

a

Answer

0

Answer

b

Question 18

The variance of a continuous probability distribution is defined as:

Accepted Answer

E[(X - μ)²]

Answer

E[X²]

Answer

E[(X - μ)]

Answer

Std. Dev. of X

Question 19

The mode of a continuous probability distribution is the value at which the:

Accepted Answer

Probability density function is maximum

Answer

Cumulative distribution function is maximum

Answer

Mean is minimum

Answer

Variance is minimum

Question 20

Which of the following is a characteristic of a continuous probability distribution?

Accepted Answer

It can take on any value within a given interval.

Answer

It only takes on a finite number of values.

Answer

It is described by a cumulative distribution function only.

Answer

The cumulative distribution function is constant.

Question 21

Which of the following is the probability density function of the standard normal distribution?

Accepted Answer

f(x) = (1 / √(2π)) * e^(-x^2 / 2)

Answer

f(x) = e^(-x^2 / 2)

Answer

f(x) = x * e^(-x^2 / 2)

Answer

f(x) = (1 / √(2π)) * e^(-x^2)

Question 22

What is the mean of the exponential distribution with parameter λ?

Accepted Answer

1 / λ

Answer

λ^2

Answer

-λ

Answer

e^(-λ)

Question 23

A random variable X has a uniform distribution on the interval [0, 1]. What is P(0.25 < X < 0.75)?

Accepted Answer

0.5

Answer

1

Answer

0.25

Answer

0.75

Question 24

Which of the following is the cumulative distribution function of the standard normal distribution?

Accepted Answer

Φ(x) = (1 / √(2π)) * ∫_{-∞}^{x} e^(-t^2 / 2) dt

Answer

Φ(x) = (1 / √(2π)) * e^(-x^2 / 2)

Answer

Φ(x) = e^(-x^2 / 2)

Question 25

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

Accepted Answer

0.6827

Answer

0.5

Answer

0.3413

Answer

0.8413

Question 26

A random variable X has a uniform distribution on the interval [a, b]. What is the probability density function of X?

Accepted Answer

f(x) = 1 / (b - a) for a ≤ x ≤ b

Answer

f(x) = e^(-(x - a) / (b - a))

Answer

f(x) = x / (b - a)

Answer

f(x) = (b - a) / x

Question 27

What is the expected value of a continuous probability distribution with probability density function f(x)?

Accepted Answer

μ = ∫_{-∞}^{∞} x * f(x) dx

Answer

μ = ∫_{0}^{1} x * f(x) dx

Answer

μ = ∫_{0}^{∞} x * f(x) dx

Answer

μ = ∫_{-∞}^{∞} f(x) dx

Question 28

Which of the following is NOT a continuous probability distribution?

Accepted Answer

Binomial

Answer

Exponential

Answer

Uniform

Answer

Normal

Question 29

The probability density function of the standard normal distribution is given by:

Accepted Answer

f(z) = (1 / √(2π)) * e^(-z²/2)

Answer

f(x) = λ * e^(-λx)

Answer

f(x) = (1 / σ√(2π)) * e^(-(x-μ)²/(2σ²))

Answer

f(x) = 1/b - a for a ≤ x ≤ b

Question 30

What is the cumulative distribution function of the exponential distribution with parameter λ?

Accepted Answer

F(x) = 1 - e^(-λx)

Answer

F(x) = λ * e^(-λx)

Answer

F(x) = e^(-λx²/2)

Answer

F(x) = (1 / λ) * e^(-λx)

Question 31

A random variable X has a uniform distribution on the interval [0, 1]. What is the probability that X is less than 0.5?

Accepted Answer

0.5

Answer

1

Answer

0.25

Answer

0.75

Question 32

Which of the following is a characteristic of the exponential distribution?

Accepted Answer

The mean is equal to the standard deviation

Answer

The distribution is symmetric

Answer

The distribution is bimodal

Question 33

Which of the following is a property of a normal distribution?

Accepted Answer

The area under the curve between any two values is equal to the probability that the random variable takes on a value in that interval.

Answer

It has a finite range.

Answer

It is a discrete distribution.

Answer

It is skewed to the right.

Question 34

The random variable X represents the height of male students in a class. If X is normally distributed with a mean of 172 cm and a standard deviation of 8 cm, what is the probability that a randomly selected male student is shorter than 160 cm?

Accepted Answer

0.0228

Answer

0.5

Answer

0.1587

Answer

0.8413

Question 35

A continuous random variable X has a uniform distribution on the interval [0, 6]. What is the expected value of X?

Accepted Answer

3

Answer

2

Answer

6

Answer

0

Question 36

The cumulative distribution function (CDF) of a continuous random variable X is given by F(x) = 0.5 + (1/4)sin(x) for 0 ≤ x ≤ π. What is the probability density function (PDF) of X?

Accepted Answer

f(x) = (1/4)cos(x) for 0 ≤ x ≤ π

Answer

f(x) = -(1/4)cos(x) for 0 ≤ x ≤ π

Answer

f(x) = - (1/4)sin(x) for 0 ≤ x ≤ π

Answer

f(x) = (1/4)sin(x) for 0 ≤ x ≤ π

Question 37

A machine produces metal rods with lengths that are normally distributed with a mean of 12 cm and a standard deviation of 1.5 cm. What is the probability that a randomly selected rod will have a length between 10 cm and 14 cm?

Accepted Answer

0.6827

Answer

0.75

Answer

0.1587

Answer

0.9545

Question 38

Which of the following statements about the standard normal distribution is TRUE?

Accepted Answer

It is symmetric about its mean.

Answer

It is positively skewed.

Answer

It has a finite range.

Answer

It is skewed to the left.

Question 39

The probability density function of a uniform distribution is given by:

Accepted Answer

f(x) = 1 / (b - a) for a ≤ x ≤ b

Answer

f(x) = x / (b - a) for a ≤ x ≤ b

Answer

f(x) = (b - x) / (b - a) for a ≤ x ≤ b

Answer

f(x) = (x - a) / (b - a) for a ≤ x ≤ b

Question 40

Which of the following statements about the cumulative distribution function of an exponential distribution is TRUE?

Accepted Answer

It is a monotonically increasing function.

Answer

It is a constant function.

Answer

It is a periodic function.

Answer

It is a monotonically decreasing function.

Question 41

Which of these is a measure of central tendency?

Accepted Answer

Mean

Answer

Variance

Answer

Standard deviation

Answer

Kurtosis

Question 42

Which of the following is a measure of data spread?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 43

What is a true statement about the normal distribution?

Accepted Answer

It is a symmetrical distribution.

Answer

It is a uniform distribution.

Answer

It is an exponential distribution.

Answer

It is a skewed distribution.

Question 44

Which statement is true about the exponential distribution?

Accepted Answer

It is a continuous distribution.

Answer

It is a normal distribution.

Answer

It is a uniform distribution.

Answer

It is a discrete distribution.

Question 45

Which of the following is a continuous probability distribution?

Accepted Answer

Normal distribution

Answer

Hypergeometric distribution

Answer

Poisson distribution

Answer

Binomial distribution

Question 46

What is the probability density function (PDF) of the normal distribution?

Accepted Answer

f(x) = (1/√(2πσ^2)) * e^(-(x-μ)^2/(2σ^2))

Answer

f(x) = 1/x

Answer

f(x) = e^(-x)

Answer

f(x) = x^2

Question 47

What is the mean of a uniform distribution with parameters 'a' and 'b'?

Accepted Answer

(a+b)/2

Answer

b

Answer

a

Answer

0

Question 48

A random variable Y follows an exponential distribution with λ = 0.5. What is the probability that Y is greater than 2?

Accepted Answer

0.3679

Answer

0.6065

Answer

0.7500

Answer

0.2500

Question 49

Which of the following is NOT a property of a probability density function?

Accepted Answer

The area under the curve is greater than 1.

Answer

The area under the curve between any two points represents the probability of the random variable taking a value between those two points.

Answer

The integral of the curve over the entire real line is equal to 1.

Answer

The curve is always non-negative.

Question 50

What is the expected value of a continuous random variable?

Accepted Answer

The mean of the random variable.

Answer

The variance of the random variable.

Answer

The mode of the random variable.

Answer

The standard deviation of the random variable.

Question 51

What is the variance of a continuous random variable?

Accepted Answer

The second moment about the mean of the random variable.

Answer

The first moment about the mean of the random variable.

Answer

The standard deviation of the random variable.