Final Exam [Bulk Review 1]

Question 1

Definition of independent events?

Answer 1

Two events, A and B, are independent if the occurrence of one does not affect the probability of the other occurring.

Question 2

P(A and B) of independent events?

Answer 2

P(AandB) = P(A) × P(B)

Question 3

Events A and B are independent if . . .

Answer 3

P(A and B) = P(A) × P(B)
P(A | B) = P(A)
P(B | A) = P(B)

Question 4

Definition of dependent events?

Answer 4

Two events, A and B, are dependent if the occurrence of one event affects the probability of the other. Commonly, P(B∣A) represents the probability of B given that A has occurred.

Question 5

P(A and B) of dependent events?

Answer 5

P(A and B) = P(B ∣ A) × P(A)

Question 6

Definition of mutually exclusive events?

Answer 6

Two events, A and B, are mutually exclusive (or disjoint) if they cannot both occur at the same time.

Question 7

P(A and B) of mutually exclusive events?

Answer 7

P(AandB) = 0

Question 8

P(AorB) of mutually exclusive events?

Answer 8

P(AorB) = P(A) + P(B)

Question 9

Definition of non-mutually exclusive events?

Answer 9

Two events, A and B, are non-mutually exclusive if they can both occur at the same time.

Question 10

P(AorB) of non-mutually exclusive events?

Answer 10

P(AorB) = P(A) + P(B) − P(AandB)

Question 11

Definition of conditional probability?

Answer 11

The probability of one event occurring given that another event has already occurred.

Question 12

P(B given that A)?

Answer 12

P(B∣A) = P(AandB) / P(A)
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Question 13

What is Baye’s Theorem for P(B∣A)?

Answer 13

P(B∣A) = P(A∣B )× P(B) / P(A)
​

Question 14

P(A given that B)?

Answer 14

P(A∣B) = P(AandB) / P(B)

Question 15

When working conditionally it is sometimes easier to calculate P(A and B) as . . .

Answer 15

P(A|B) ⋅ P(B) OR P(B|A) ⋅ P(A)

Question 16

Can events be independent and mutually exclusive?

Answer 16

NO unless one of the events has a zero probability (i.e., one of the events cannot occur). This is because mutually exclusive events can never occur together, whereas independent events can.

Question 17

When asked for P(A or B) of independent events?

Answer 17

P(AorB) = P(A) + P(B) − P(AandB)
Because independent events are NOT mutually exclusive

Question 18

Addition Rule for . . .
Mutually Exclusive Events:
Non-Mutually Exclusive Events:

Answer 18

Mutually Exclusive Events:
P(AorB) = P(A) + P(B)

Non-Mutually Exclusive Events:
P(AorB) = P(A) + P(B) − P(AandB)

Question 19

Multiplication Rule for . . .
Independent Events:
Dependent Events:

Answer 19

Independent Events:
P(AandB) = P(A) × P(B)

Dependent Events:
P(AandB) = P(A) × P(B∣A)

Question 20

If S is the sample space for any event A⊆S . . .
A ⋂ Ac = Ø, they are _________
A ⋃ Ac = S, they are _________

Answer 20

disjoint, partitions

Question 21

Demorgan’s Law

Answer 21

(A ⋃ B)c = Ac ⋂ Bc
(A ⋂ B)c = Ac ⋃ Bc

Question 22

Inclusion-Exclusion Principle

Answer 22

P(A ⋃ B) = P(A) + P(B) - P(A ⋂ B)

Question 23

If A and B are disjoint events then,
P(A ⋃ B) = _____
P(A) = _______
P(A ⋂ B) = ____

Answer 23

P(A) + P(B)
1 - P(Ac)
Ø

Question 24

Events A and B are independent if . . .

Answer 24

P(A ⋂ B) = P(A) ⋅ P(B)
P(A | B) = P(A)
P(B | A) = P(B)

Question 25

Conditional Probability

Answer 25

P(A | B) = P(A ⋂ B)/ P(B)

Question 26

When working conditionally, P(A ⋂ B) = ________

Answer 26

P(A | B) ⋅ P(B) OR P(B | A) ⋅ P(A)

Question 27

Baye's Theorem

Answer 27

P(A | B) = [P(B | A) ⋅ P(A)] / P(B)

Question 28

For mutually exclusive events A and B, P(A ⋂ B) = ____

Answer 28

0

Question 29

For independent events A and B, P(A ⋃ B) = _________

Answer 29

P(A) + P(B) - P(A and B)

Question 30

If A and B are not disjoint then, P(A ⋃ B) = _________

Answer 30

P(A) + P(B) - P(A ⋂ B)

Question 31

How do you determine if events are mutually exclusive?

Answer 31

P(A ⋂ B) = 0

Question 32

For mutually exclusive events A and B, P(A ⋃ B) = _________

Answer 32

P(A) + P(B)

Question 33

Mutually exclusive is also known as ____

Answer 33

disjoint

Question 34

The source of the randomness in a random variable is _______________, in which a sample outcome s ∈ S is chosen according to a ___________

Answer 34

the experiment itself probability function P

Question 35

The **probability mass function (PMF)** of a discrete r.v. X is the function pX given by _________. Note that this is ________ if x is in the support of X, and _________ otherwise.

Answer 35

pX (x) = P(X = x) positive, zero

Question 36

The **cumulative distribution function (CDF)** of an r.v. X is the function F(X) given by __________

Answer 36

F(x) = P(X ≤ x)

Question 37

The expected value of a discrete r.v. X whose distinct possible values are x1, x2, . . ., is defined by . . .

Answer 37

Question 38

For any r.v.s X, Y and any constant c there are 4 primary manipulations to know for the E(X): 1. E(c) = 2. E(X + Y) = 3. E(X + c) = 4. E(cX) =

Answer 38

Question 39

The variance of an r.v. X is ________

Answer 39

Question 40

For any r.v.s X, Y and any constant c there are 4 primary manipulations to know for the Var(X): 1. Var(X + c) = 2. Var(cX) = 3. Var(X + Y) = 4. Var(X) >= 0 when . . .

Answer 40

Question 41

An experiment that can result in either a "success" or a "failure" (but not both) is called a ___________

Answer 41

Bernoulli trial

Question 42

Suppose that n independent Bernoulli trials are performed, each with the same success probability p. Let X be the number of successes. The distribution of X is called the ________ distribution

Answer 42

Binomial

Question 43

Consider a sequence of independent Bernoulli trials, each with the same success probability p ∈ (0, 1), with trials performed until a success occurs. Let X be the number of failures **before** the first successful trial. Then X has the_____________

Answer 43

Geometric distribution

Question 44

An r.v. X has the __________ with parameter λ if it explains the distribution of . . . 1. the number of successes in a particular region or _________ 2. a large number of trials, each with a ___________.

Answer 44

Poisson distribution interval of time small probability of success

Question 45

The Poisson paradigm is also called the law of rare events. The interpretation of "rare" is that the _____ are small, not that _______ is small.

Answer 45

p λ

Question 46

An r.v. has a continuous distribution if its CDF is ___________. We also allow there to be endpoints (or finitely many points) where the CDF is continuous but not differentiable, as long as . . .

Answer 46

differentiable the CDF is differentiable everywhere else

Question 47

Let X be a continuous r.v. with PDF f . Then the CDF of X is given by . . .

Answer 47

Question 48

Probabilities for continuous random variables are specified by__________. This leads us to the special rule that P(X = x) = _______ for continuous random variables

Answer 48

the area under a curve 0

Question 49

For sets of the form [a, b], (a, b], [a, b), (a, b), the CDF is dereived from the PMF by . . .

Answer 49

Question 50

To get a desired probability for a continuous r.v. you . . .

Answer 50

integrate the PDF over the desired range

Question 51

The PDF f of a continuous r.v. must satisfy the following two criteria:

Answer 51

Question 52

The expected value (also called the expectation or mean) of a continuous r.v. X with PDF f is . . .

Answer 52

Question 53

What is the power rule for differentiating a function?

Answer 53

Question 54

What is the power rule for integrating a function?

Answer 54

Question 55

The CDF of the continuous uniform is . . .

Answer 55

Question 56

A continuous r.v. X is said to have the Exponential distribution with parameter λ > 0 if its CDF is . . .

Answer 56

Question 57

A continuous distribution is said to have the memoryless property if a random variable X from that distribution satisfies . . .

Answer 57

Question 58

the **central limit theorem** says that under very weak assumptions, the sum of a large number of i.i.d. random variables has an . . .

Answer 58

approximately Normal distribution, regardless of the distribution of the individual r.v.s.

Question 59

What are the three important symmetry properties that can be deduced from the standard Normal PDF and CDF?

Answer 59

Question 60

For X ∼ N(μ, σ2) the standardized version of X is given by . . .

Answer 60

Question 61

If X ∼ N(μ, σ2) then . . . 1. P(|X − μ| < σ) ≈ 2. P(|X − μ| < 2σ) ≈ 3. P(|X − μ| < 3σ) ≈ This is commonly known as _____ rule

Answer 61

1. P(|X − μ| < σ) ≈ 0.68 2. P(|X − μ| < 2σ) ≈ 0.95 3. P(|X − μ| < 3σ) ≈ 0.997 68-95-99.7% rule

Question 62

What is P(c < x < d) for X~Unif(a,b)?

Answer 62

(d-c/b-a)

Question 63

How do you find percentile of a normal distribution?

Answer 63

X = μ + (z x σ) where z corresponds to the Z score (X − μ /σ) of the desired percentile