Ch 2

Question 1

What is conditional probability?

Answer 1

Let A, B ⊆ Ω be events with P (A)̸ /= 0. Then the conditional probability of B given A is defined by
P (B|A) = P (B ∩ A)/P (A)

Question 2

Give another form of conditional probability equation.

Answer 2

P (A ∩ B) = P (A|B) · P (B)

Question 3

What is the union of events Ai where i€{1,…,n} where the union=\0?

Answer 3

P (A1 ∩ . . . ∩ An ) = P (A1) · P (A2|A1) · . . . · P (An |A1 ∩ . . . ∩ An−1)

Question 4

What is the law that of total probability?

Answer 4

Suppose we partition a sample space Ω into n pairwise disjoint events A1, . . . , An with P (Ai ) > 0 for all i =
1, . . . , n. Let B ⊆ Ω be an event. Then
P (B) = P (B|A1) P (A1) + · · · + P (B|An ) P (An )

Question 5

Using law of total probability what is P(B) in case of just B,A,A^c?

Answer 5

P(B)=P(B|A)P(A)+P(B|A^c)P(A^c)

Question 6

What is bayes formula?

Answer 6

Let A, B be events with P (B)̸ /= 0. Then
P (A|B) = P (B|A) P (A)/P (B)

Question 7

What is bayes theorem?

Answer 7

Let our sample space Ω be partitioned into disjoint events Ak , k = 1, 2, . . . , n. Then for 1 ≤ j ≤ n,
P(Aj |B)= P (B|A j)P(A j)/
Sigma n
k=1P (B|Ak ) P (Ak)

Question 8

When are two events independent?

Answer 8

When P (A ∩ B) = P (A) P (B)

Question 9

Assuming P(B)=\0. What is A and B being independent the same as saying?

Answer 9

P(A)=P(A|B) or the probability of A does not change regardless of whether B happened or not

Question 10

What is a random variable?

Answer 10

A measurable function on the sample space

Question 11

What is a cumulative distribution function(cdf)?

Answer 11

When X is a random variable,
FX (x) := P (X ≤ x)

Question 12

Explain what the cdf actually does.

Answer 12

a function that describes the cumulative probability that a random variable takes on a value less than or equal to a particular value

Question 13

Describe the properties of a cdf.

Answer 13

•non decreasing: as x increases F(x) does not decrease
•limx->-inf F(x)=0 and limx->inf F(x)=1
•right continuous: for any x, F(x) approaches F(x) from the right

Question 14

What is a discrete random variable?

Answer 14

If for a random variable X there exists a set {xi : i ∈ I } for a finite or at at most countably infinite index set I ,
such that sigma i ∈I P (X = xi ) = 1, then we call X a discrete random variable.
This means that we call a random variable discrete if it can take at most countably many values

Question 15

How do we define a probability mass function?

Answer 15

For a discrete random variable X , we define its probability mass function p by
p(x j ) = P (X = x j)
for all values x j that X can attain. Quite often we will write p j := p(x j )

Question 16

Explain what a pmf actually is?

Answer 16

Afunction that gives the probability that a discrete random variable takes on a specific value. For a discrete random variable X, the PMF, denoted as
P(X=x), provides the probability that
X is equal to some particular value
x

Question 17

What are the properties of a pmf?

Answer 17

•Non-negative: The PMF always outputs probabilities that are non-negative for all values of x: P(X=x)≥0forallx
•Probabilities sum to 1: The sum of the probabilities of all possible values of
X must equal 1, as one of the values must occur:∑P(X=x)=1
•Specific to discrete variables: The PMF applies only to discrete random variables, as it assigns probabilities to distinct outcomes.

Question 18

Explain probability density function(pdf).

Answer 18

A function that describes the likelihood of a continuous random variable taking on a specific value within an interval. Unlike a probability mass function (PMF), which applies to discrete random variables, a PDF applies to continuous random variables, which can take any value in a continuous range.

Question 19

Definition of continuous random variables and pdf.

Answer 19

Let X : Ω → R be a random variable. If there exists a (Lebesgue) integrable function fX : R → [0, ∞) such that
for every “nice” set B ⊂ R we have
P (X ∈ B) =integralB fX (x)d x,
then we call X a continuous random variable and fX the probability density function (pdf) of X

Question 20

What are the properties of a pdf?

Answer 20

•Non-negative: The PDF is always non-negative for all values of the random variable x:f X(x)≥0forallx
•Area under the curve equals 1: The total area under the curve of the PDF across all possible values of the random variable must equal 1. This reflects the fact that the total probability for all possible outcomes is 1:∫ −∞∞f X(x)dx=1
•The probability of a specific value is 0: For a continuous random variable, the probability of the variable taking any specific single value (e.g.,P(X=x)) is 0. Instead, probabilities are determined over intervals. For example, the probability that the variable falls within a range [a,b] is calculated as the integral of the PDF over that range:P(a≤X≤b)=∫ ab f X(x)dx
•Relationship to the CDF: The PDF is the derivative of the Cumulative Distribution Function (CDF). Conversely, the CDF is the integral of the PDF:F X (x)=∫ −∞->x
f X(t)dt