Chapter 4

Question 1

What is a continuous random variable?

Answer 1

A variable that can take any value within a given range.

Question 2

What is the key difference between discrete and continuous random variables?

Answer 2

Discrete random variables have countable outcomes, while continuous random variables have uncountable outcomes.

Question 3

Why does P(X = x) = 0 for continuous random variables?

Answer 3

Because there are infinitely many possible values, making the probability of any single value zero.

Question 4

What is the cumulative distribution function (CDF)?

Answer 4

A function that gives the probability that a random variable is less than or equal to a given value.

Question 5

What is the probability density function (PDF)?

Answer 5

A function that describes the likelihood of a continuous random variable taking on a specific value.

Question 6

What is the relationship between the CDF and PDF?

Answer 6

The PDF is the derivative of the CDF.

Question 7

What condition must a PDF satisfy?

Answer 7

It must be non-negative and integrate to 1 over its entire range.

Question 8

How do you compute the probability of a range of values for a continuous random variable?

Answer 8

By integrating the PDF over that range.

Question 9

What is the expected value of a continuous random variable?

Answer 9

The integral of x times the PDF over all possible values.

Question 10

What is the formula for expectation in continuous variables?

Answer 10

E(X) = ∫ x fX(x) dx.

Question 11

What is variance in continuous probability distributions?

Answer 11

A measure of how spread out the values of a random variable are.

Question 12

What is the formula for variance?

Answer 12

Var(X) = E(X²) - (E(X))².

Question 13

What is the uniform distribution?

Answer 13

A distribution where all values in an interval are equally likely.

Question 14

What is the PDF of the uniform distribution?

Answer 14

fX(x) = 1 / (b - a) for a ≤ x ≤ b, otherwise 0.

Question 15

What is the expected value of a uniform distribution?

Answer 15

E(X) = (a + b) / 2.

Question 16

What is the variance of a uniform distribution?

Answer 16

Var(X) = (b - a)² / 12.

Question 17

What is the exponential distribution used for?

Answer 17

Modeling the time between independent events.

Question 18

What is the PDF of the exponential distribution?

Answer 18

fX(x) = λ e^(-λx) for x ≥ 0, otherwise 0.

Question 19

What is the expected value of an exponential distribution?

Answer 19

E(X) = 1 / λ.

Question 20

What is the variance of an exponential distribution?

Answer 20

Var(X) = 1 / λ².

Question 21

What is the memoryless property of the exponential distribution?

Answer 21

P(X > x + t | X > t) = P(X > x), meaning past waiting time does not affect future probabilities.

Question 22

What is the normal (Gaussian) distribution?

Answer 22

A continuous probability distribution that is symmetric around its mean.

Question 23

What is the PDF of a normal distribution?

Answer 23

fX(x) = (1 / (σ√(2π))) e^(-(x - μ)² / (2σ²)).

Question 24

What are the parameters of a normal distribution?

Answer 24

Mean (μ) and standard deviation (σ).

Question 25

What is the standard normal distribution?

Answer 25

A normal distribution with mean 0 and variance 1.

Question 26

What is the CDF of the normal distribution called?

Answer 26

The standard normal cumulative distribution function (Φ).

Question 27

Why is the normal distribution important?

Answer 27

Because of the Central Limit Theorem, which states that sums of many independent random variables tend to follow a normal distribution.

Question 28

What is a linear transformation of a normal variable?

Answer 28

If X ~ N(μ, σ²), then Y = aX + b is also normally distributed.

Question 29

How is the probability of a normal variable computed?

Answer 29

Using the standard normal table or numerical integration.

Question 30

What is the key takeaway about continuous distributions?

Answer 30

They are described by PDFs and CDFs, and probabilities are computed via integration.