12 - Probability Distributions

Question 1

What are the two types of probability distributions?

Answer 1

Discrete and Continuous

Question 2

For a discrete random variable X, what does P (X = x) refer to?

Answer 2

The probability that X is equal to a particular value of x.

Question 3

What must the sum of all probabilities in any probability distribution equal?

Answer 3

1

Question 4

Give the formula for the expectation (mean) of a discrete random variable.

Answer 4

𝐸(𝑋) = ΣxP(X=x)

Question 5

Give the formula for the variance of a discrete random variable.

Answer 5

𝑉𝑎𝑟(𝑋) = Σx²P(X=x) - E(X)²

Question 6

What are the formulas for linear coding of data?

Answer 6

𝐸 (𝑎𝑋 + 𝑏) = aE(X) + b and 𝑉𝑎𝑟 (𝑎𝑋 + 𝑏) = a²Var(X)

Question 7

Name two common examples of discrete distributions mentioned.

Answer 7

Discrete Uniform Distribution and Binomial Distribution

Question 8

Describe the properties of a discrete uniform distribution.

Answer 8

A discrete random variable X is defined over a set of n outcomes, and each outcome is equally likely.

Question 9

Give the formulas for the expectation and variance of a discrete uniform distribution.

Answer 9

𝐸(𝑋) = (n+1)/2 and 𝑉𝑎𝑟(𝑋) = (n²-1)/12

Question 10

What is a common example of a continuous distribution, and what is another name for it?

Answer 10

Continuous Uniform Distribution, also known as the Rectangular Distribution.

Question 11

Give the formulas for the expectation and variance of a continuous uniform distribution over the interval [a, b].

Answer 11

𝐸(𝑋) = (a+b)/2 and 𝑉𝑎𝑟(𝑋) = (b-a) ²/12

Question 12

If 𝑋1~𝑁 (𝜇1, 𝜎1 2) and 𝑋2~𝑁 (𝜇2, 𝜎2 2) are independent, what is 𝐸 (𝑋1 ± 𝑋2)?

Answer 12

𝐸(𝑋1) ± 𝐸(𝑋2) = 𝜇1 ± 𝜇2

Question 13

If 𝑋1~𝑁 (𝜇1, 𝜎1 2) and 𝑋2~𝑁 (𝜇2, 𝜎2 2) are independent, what is 𝑉𝑎𝑟 (𝑋1 ± 𝑋2)?

Answer 13

𝑉𝑎𝑟(𝑋1) + 𝑉𝑎𝑟(𝑋2) = 𝜎1 2 + 𝜎2 2

Question 14

If 𝑿~𝑵 (𝝁𝟏, 𝝈𝟏 𝟐) and 𝒀~𝑵 (𝝁𝟐, 𝝈𝟐 𝟐) are independent, what is the distribution of 𝒂𝑿 ± 𝒃𝒀?

Answer 14

𝒂𝑿 ± 𝒃𝒀~𝑵(𝒂𝝁𝟏 ± 𝒃𝝁𝟐, 𝒂𝟐𝝈𝟏 𝟐 + 𝒃𝟐𝝈𝟐 𝟐)