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Probability > Probability > Flashcards

Probability Flashcards

1
Q

What is a finite sample space?

A

Finite sample space has a finite number of outcomes/elements.

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2
Q

What is a countably infinite sample space?

A

Countably infinite sample space is a sample space which outecomes can be put into to a one-to-one correspondence with the set of natural numbers.

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3
Q

What is a discrete sample space?

A

If a sample space is finite or countably infinite it can also be called discrete.

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4
Q

Another name for a discrete sample space.

A

Countable

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5
Q

What is a continuous sample space?

A

Continuous sample space contains outcomes that can take any value of any real number on a certain interval.

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6
Q

What are de Morgans laws?

A

(A ∩ B)^c = A^c U B^c.

(A U B)^c = A^c ∩ B^c.

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7
Q

What is a sure event?

A

Set S - which is the whole sample space.

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8
Q

What is a null event?

A

Empty set

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9
Q

What is an elementary event?

A

One outcome in the sample space/experiment.

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10
Q

What are mutually exclusive events? (set notation)

A

A∩ B = ∅

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11
Q

What is a relative frequency?

A 
If n(x) represents the number of times event x has occurred in the total number of experiment trials  N, then 
 fx= n(x) / N
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12
Q

What is statistical regularity?

A

P(x) = lim fx = lim n(x) / N (n is goingt to infinity)

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13
Q

What is the primary objective of probability modelling?

A

Given a sample space S, we aim to assign a real number P(A) to each possible outcome of A, which describes the likelihood that A will occur if the experiment is performed.

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14
Q

What is an event?

A

An event is a subset of a sample space.

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15
Q

How to think of P(A) as a function? What is the domain? Range?

A

All possible outcomes are the domain and the probabilities are the range.

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16
Q

What are three axioms of probability?

A
  1. 0<=P(A)<=1
  2. P(S) = 1
  3. P ( U Ai) =P(A1)UP(A2)U…UP(Ai) = P(A1)+P(A2)+…P(Ai)
    If A1, A2 are pairwise mutually exclusive events.
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17
Q

What is a classical probability model?

A

Implies that each elementary event has equal probability

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18
Q

What is an object chose of random?

A

If an object is chosen at random it means it had the same probability of being picked as all the other object in the sample space.

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19
Q

P(A U B U C) ?

A

P( A ) + P (B) + P(C) - P(A ∩ B)- P(A ∩ C) - P(C ∩ B) + P (A ∩ B ∩ C)

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20
Q

If A is a subset of B, then P(A) ? P(B)

< / > / =

A

<=

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21
Q

What is the formula for conditional probability? Explain

A

P(A|B) = P(A ∩ B) / P(B)
Conditional probability determines a probability of an event on the reduced sample space.

P(A|B) determines how likely event A, given that B has occurred. As usual, we can look what is the total number of events when A and B have occurred and divide it by the total number of events B. Dividing the numerator and denominator by the total sample size allows determining the conditional prob by unconditional prob.

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22
Q

What is the total law of probability?

A

P(A) = P(A1|B)P(B) + P(A2|B)P(B) + … + P(An|B)P(B)

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23
Q

What is the Multiplication law of probability?

A

P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A)

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24
Q

What is a marginal probability?

A

Basically anytime you are in interested in a single event irrespective of any other event (i.e. “marginalizing the other event”), then it is a marginal probability.

25
Q

Which two events are called independent?

A

A and B are independent if P(A ∩ B) = P(A) * P(B)

26
Q

What is Bayes’ Rule?

A

P(B|A) = P(A|B)*P(B) / P(A)

27
Q

Which k events are independent / mutually independent?

A

P(A1 ∩ A2 ∩ An) = P(A1) * P(A2)*P(An)

28
Q

What is a random variable?

A

Random variable is a function defined over the sample space S, that assigns a real number X(e) = x

29
Q

What is a discrete random variable?

A

If the set of all possible values of a random var X is a countable set, then X is a discrete random variables.

30
Q

What is the probability density function of a discrete random variable?

A

PDF is f(x) = P [ X = x]
It assigns the probability of each possible value x.

A function is a pdf if and only if f(xi)>=0 and sum over all x f(xi) = 1

31
Q

What is another name of pdf?

A

Probability mass function

32
Q

What is a cumulative distribution function?

A

F(x) = P [X<=x]

33
Q

A function is a cdf if and only if what? (4)

A

lim x -> - inf F(x) = 0
lim x ->1 F(x) = 1
lim h -> 0+ F(x+h) = F(x) (right continious)
a<b> F(a) <= F(b)</b>

34
Q

What is the general relationship between F(x) and f(x)?

A 
f(x1) = F ( x1 ) 
f(xi) = F ( xi ) - F ( xi-1 )
35
Q

What is the expected value of a discrete random variable X?

A

If pdf f(x), then E ( X) = (Sum over all xi) f(xi)*xi

36
Q

What are other names for the expected value?

A

Mean

Expectation

37
Q

What is a continuous random variable?

A

Variable can be considered continious if there is a pdf function of X, such that the CDF is
F(x) = x to (-inf) ∫ f(t)dt

38
Q

With a continuous random variable how to get from CDF to pdf?

A

f(x) = F’(x)

39
Q

With a continuous random variable how to get from pdf to CDF?

A

F(x) = x to (-inf) ∫ f(t)dt

40
Q

Consider events of a continuous random variable of the form [X ∈ I], where I is an interval. What is important to remember?

A

The probability of the event is the same weather I include the endpoints or not.

41
Q

When is a function f(x) a pdf for some continuous random variable X?

A

If and only if
f(x) > = 0
and for all real x:
(inf) to (-inf) ∫ f(x)dx = 1

42
Q

How to find a P[a

A

P[a

43
Q

What is the expected value of a continuous random variable?

A

E(X) = (inf) to (-inf) ∫ x * f(x)dx

44
Q

What are other notations for E(X)?

A

μ , μx,
Mean / expectation
Center of mass

45
Q

What does a percentile p of a distribution of a continuous random variable do?

A

It indicates the value below which a given percentage of observations in a group of observations fall.

F(x) = p

46
Q

What is a median of a distribution?

A

50th percentile

47
Q

What is a mode of a distribution?

A

If there is a uniques maximum of pdf at x = m0, such as max f(x) = f( m0), then m0 is a mode.

48
Q

When is a distribution symmetric about c?

A

Distribution symmetric about c if f(c-x) = f(c+x) for all x.

49
Q

What is another name for a skewed distribution?

A

Asymmetric.

50
Q

Formula for the variance of X

A

var(x) = E[ (X-μ )^2 ]
or
E[ (X^2)] +E(X)^2 = E[ (X^2)] + μ^2

51
Q

A formula for the variance of X (2)

A

var(x) = E[ (X-μ )^2 ]
or
E[ (X^2)] +E(X)^2 = E[ (X^2)] + μ^2

52
Q

Kth moment about the mean

A

μk = E(X-μ)^k

53
Q

Var(aX+b) = ?

A

a^2 * Var(X)

54
Q

What can you say about the third moment about mean of X if the distribution is symmetric?

A

μ3 = 0

55
Q

What is Markov inequality?

What is the proof for Markov inequality?

A

P (|X|> = c) = E[X] / c

Proof: see page 76

56
Q

What is the Chebishev inequality?

A

P(|X - μ| >= kσ) <= 1 / K^2
and
P(|X - μ| <= kσ) >=1 - 1 / K^2

57
Q

What is the Chebishev inequality?

A

P(|X - μ| >= kσ) <= 1 / K^2
and
P(|X - μ| <= kσ) >=1 - 1 / K^2

58
Q

Degenerate distribution

A

Distribution with σ^2= 0 and distribution . concentrated at one point.

59
Q

What is a moment of a function?

A

The moments are special expected values, which include the mean and variance as particular cases, and also provide descriptive measures for other characteristics
such as skewness of a distribution.

Probability (1 decks)

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