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STAT 30133 > Final Exam [Bulk Review 1] > Flashcards

Final Exam [Bulk Review 1] Flashcards

Exam concepts 1 (Axioms of Probability) through 4 (Distributions)

1
Q

Definition of independent events?

A

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

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2
Q

P(A and B) of independent events?

A

P(AandB) = P(A) × P(B)

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3
Q

Events A and B are independent if . . .

A

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

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4
Q

Definition of dependent events?

A

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.

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5
Q

P(A and B) of dependent events?

A

P(A and B) = P(B ∣ A) × P(A)

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6
Q

Definition of mutually exclusive events?

A

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

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7
Q

P(A and B) of mutually exclusive events?

A

P(AandB) = 0

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8
Q

P(AorB) of mutually exclusive events?

A

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

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9
Q

Definition of non-mutually exclusive events?

A

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

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10
Q

P(AorB) of non-mutually exclusive events?

A

P(AorB) = P(A) + P(B) − P(AandB)

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11
Q

Definition of conditional probability?

A

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

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12
Q

P(B given that A)?

A

P(B∣A) = P(AandB) / P(A)
​

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13
Q

What is Baye’s Theorem for P(B∣A)?

A

P(B∣A) = P(A∣B )× P(B) / P(A)
​

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14
Q

P(A given that B)?

A

P(A∣B) = P(AandB) / P(B)

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15
Q

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

A

P(A|B) ⋅ P(B) OR P(B|A) ⋅ P(A)

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16
Q

Can events be independent and mutually exclusive?

A

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.

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17
Q

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

A

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

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18
Q

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

A

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

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

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19
Q

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

A

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

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

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20
Q

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

A

disjoint, partitions

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21
Q

Demorgan’s Law

A

(A ⋃ B)c = Ac ⋂ Bc
(A ⋂ B)c = Ac ⋃ Bc

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22
Q

Inclusion-Exclusion Principle

A

P(A ⋃ B) = P(A) + P(B) - P(A ⋂ B)

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23
Q

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

A

P(A) + P(B)
1 - P(Ac)
Ø

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24
Q

Events A and B are independent if . . .

A

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

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25
Q

Conditional Probability

A

P(A | B) = P(A ⋂ B)/ P(B)

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26
Q

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

A

P(A | B) ⋅ P(B) OR
P(B | A) ⋅ P(A)

27
Q

Baye’s Theorem

A

P(A | B) = [P(B | A) ⋅ P(A)] / P(B)

28
Q

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

A

0

29
Q

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

A

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

30
Q

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

A

P(A) + P(B) - P(A ⋂ B)

31
Q

How do you determine if events are mutually exclusive?

A

P(A ⋂ B) = 0

32
Q

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

A

P(A) + P(B)

33
Q

Mutually exclusive is also known as ____

A

disjoint

34
Q

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

A

the experiment itself
probability function P

35
Q

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.

A

pX (x) = P(X = x)

positive, zero

36
Q

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

A

F(x) = P(X ≤ x)

37
Q

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

A
38
Q

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) =

A
39
Q

The variance of an r.v. X is ________

A
40
Q

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 . . .

A
41
Q

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

A

Bernoulli trial

42
Q

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

A

Binomial

43
Q

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_____________

A

Geometric distribution

44
Q

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 ___________.

A

Poisson distribution
interval of time
small probability of success

45
Q

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

A

p
λ

46
Q

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 . . .

A

differentiable
the CDF is differentiable everywhere else

47
Q

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

A
48
Q

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

A

the area under a curve
0

49
Q

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

A
50
Q

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

A

integrate the PDF over the desired range

51
Q

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

A
52
Q

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

A
53
Q

What is the power rule for differentiating a function?

A
54
Q

What is the power rule for integrating a function?

A
55
Q

The CDF of the continuous uniform is . . .

A
56
Q

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

A
57
Q

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

A
58
Q

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

A

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

59
Q

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

A
60
Q

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

A
61
Q

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

A
  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
62
Q

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

A

(d-c/b-a)

63
Q

How do you find percentile of a normal distribution?

A

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

STAT 30133 (11 decks)

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