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Mathematics > Probability & Statistics > Flashcards

Probability & Statistics Flashcards

1
Q

Definition of a Sample Space

A

The sample space is the set of all possible outcomes for the experiment. It is denoted by S

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2
Q

Definition of an Event

A

An event is a subset of the sample space. The event occurs if the actual outcome is an element of this subset.

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3
Q

Definition of a Simple Event

A

An event is a simple event if it consists of a single element of the sample space S

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4
Q

Meaning of Disjoint events

A

We say two sets A and B are disjoint if they have no element in common, i.e, A ∩ B = {}

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5
Q

De Morgan’s laws

A

(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc

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6
Q

Kolgromov’s axioms for probability

A

a) For every event A we have P(A) >= 0,

b) P(S) = 1,

c) If A1, A2, …, An are n pairwise disjoint events
then
P(A1 U A2 U … U An) = P(A1) + P(A2) + … + P(An)

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7
Q

Complement Rule for Probability

A

If A is an event then
P(Ac) = 1 - P(A)

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8
Q

Probability of an Empty Set

A

P(∅) = 0

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9
Q

Probability of an Event Upper Bound

A

If A is an event then P(A) <= 1

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10
Q

Probability of a Subset

A

If A and B are events and A ⊆ B then
P(A) <= P(B)

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11
Q

Probability of a Finite Event

A

The probability of a finite set is the sum of the probabilities of the corresponding simple events.

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12
Q

Inclusion-exclusion for two events

A

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

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13
Q

Inclusion-exclusion for three events

A

P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)

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14
Q

Ordered with replacement (repetition allowed)

A

n^r

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15
Q

Ordered without replacement (no repetition)

A

n!
(n−r)!

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16
Q

Conditional Probability

A

If E1 and E2 are events and P(E1) /= 0 then the conditional probability of E2 given E1, usually denoted by P(E2|E1), is

P(E2|E1) = P(E1 ∩ E2)
P(E1)

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17
Q

Unordered without replacement (no repetition)

A

nCr (n)
(r)

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18
Q

Defenition of Independence

A

We say that the events E1 and E2 are (pairwise) independent if

P(E1 ∩ E2) = P(E1)P(E2)

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19
Q

When are three events E1, E2, and E3 called pairwise independent

A

P(E1 ∩ E2) = P(E1)P(E2),
P(E1 ∩ E3) = P(E1)P(E3),
P(E2 ∩ E3) = P(E2)P(E3).

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20
Q

When are three events E1, E2, and E3 called mutually independent

A

P(E1 ∩ E2 ∩ E3) = P(E1)P(E2)P(E3)

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21
Q

When are two events E1 and E2 are said to be conditionally independent given an event E3

A

P(E1 ∩ E2|E3) = P(E1|E3)P(E2|E3)

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22
Q

Definition of Random Variable

A

A random variable is a function from S to R

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23
Q

Definition of Discrete Random Variables

A

A random variable X is discrete if the set of values that X takes
is either finite or countably infinite.

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24
Q

Definition of Probability Mass Functions

A

The probability mass function (p.m.f.) of a discrete random
variable X is the function which given input x has output P(X = x)

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25
Q

Sum of Probabilities for a Discrete Random Variable

A

The sum of the Outputs must equal 1

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26
Q

Definition of Expectation

A

If X is a discrete random variable which takes values x1, x2, x3, . . ., then the expectation of X (or the expected value of X) is defined by
E(X) = x1P(X = x1) + x2P(X = x2) + x3P(X = x3) + · · · .

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27
Q

Bound on Expectations of a Random Variable

A

m ≤ E(X) ≤ M

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28
Q

Expectation of a function

A

E( f(X) ) = f(x1)P(X = x1) + f(x2)P(X = x2) + f(x3)P(X = x3) + · · ·

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29
Q

Definition of Moments

A

The nth moment of the random variable X is the expectation E(X^n)

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30
Q

Definition of Variance

A

Var(X) = [x1 − E(X)]2P(X = x1) + [x2 − E(X)]2P(X = x2)
+ [x3 − E(X)]2P(X = x3) + …

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31
Q

Variance formula

A

Var(X) = E(X^2) − [E(X)]^2

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32
Q

Linear function of expectation

A

E(aX + b) = aE(X) + b

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33
Q

Linear function of variance

A

Var(aX + b) = a^2Var(X)

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34
Q

What is a Bernoulli(p) distribution:

A

It is where a random variable X only takes values 0 and 1

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35
Q

Bernoulli distribution Expectation and Variance

A

E(X) = p, Var(X) = p(1 − p)

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36
Q

What is Binomial distribution:

A

A discrete random variable X has the Binomial(n, p) distribution, denoted X ∼ Bin(n, p), if its p.m.f. is :

P(x =k) = nCk x p^k x (1-p)^n-k

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37
Q

Binomial distribution Expectation and Variance

A

E(X) = np, Var(X) = np(1 − p)

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38
Q

What is Geometric distribution:

A

A discrete random variable X has the Geometric(p) distribution, denoted X ∼ Geom(p), if its p.m.f. is :

P(X = k) = p(1 − p)^k−1

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39
Q

Geometric distribution Expectation and Variance

A

E(X) = 1/p
Var(X) = 1 − p/ p^2

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40
Q

What is Hypergeometric distribution

A

the hypergeometric distribution describes the probability of successes in draws, without replacement, from a finite population

P(X = k) = mCk x (n-m)C(l-k) / nCl

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41
Q

Hypergeometric distribution Expectation and Variance

A

E(X) = l x (m/n)
Var(X) = l x (m/n) x (n-m/n) x (n-l/n-1)

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42
Q

What is Negative binomial distribution

A

The negative binomial distribution models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified number of successes occurs

P(X = k) = (k+r-1)C(r-1) x p^r x (1-p)^k

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43
Q

Negative Binomial distribution Expectation and Variance

A

E(T) = (1-p)r/p
Var(T) = (1-p)r/p^2

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44
Q

What is Uniform distribution

A

uniform distribution refers to a type of probability distribution in which all outcomes are equally likely

P(X=k) = 1/(n+1) if m<=k<=n+m

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45
Q

(discrete) Uniform distribution Expectation and Variance

A

where n = b-a, and m=a
E(X) = m +(n/2)
Variance = n(n+2)/12

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46
Q

What is Poisson distribution

A

Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate.

P(X=k) = (λ^k/k!) x e^-λ

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47
Q

Poisson distribution Expectation and Variance

A

E(X) = λ, Var(X) = λ

48
Q

Cumulative distribution function

A

The cumulative distribution function (c.d.f.) of a random variable X is the function which given t has output
P(X ≤ t).

49
Q

Moment Generating function

A

Let X be a discrete random variable
which takes integer values. The moment generating function (mgf ) of X is the function which given t has output E(e^(tX))

50
Q

Definition of a Continuous random variable

A

We say that a random variable X is a continuous random variable if there exists a
continuous function fx from R to [0, ∞) with the following property:

P(a<= X <= b) = integral of fx(t)

51
Q

Expectation and Variance of crv

A

E(X) = integral tfx(dt)
Var(X) = E(X^2) − (E(X))^2

52
Q

(continuous) Uniform Expectation and Variance

A

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

53
Q

Exponential distribution

A

It is often used to model the time elapsed between events

P(X=t) = λe^λt

54
Q

Exponential distribution Expectation and Variance

A

E(X) = 1/λ
Var(X) = 1/λ^2

55
Q

Joint Probability mass function

A

Let X and Y be two discrete random variables defined on the same sample space and taking values x1, x2, . . . and y1, y2, . . . respectively. The
function:

(xk, yl) → P( (X = xk) ∩ (Y = yl) )

56
Q

Marginal Probability

A

P(X=xk) = suml P(X = xk, Y = yl)

same goes the other way

the idea is that if we only care about the probability of X
taking a particular value, we need to sum over all possible values of Y

57
Q

Expectations of 2 Variables

A

If g(X, Y ) is a real-valued function of the two discrete random variables X and Y then the expectation of g(X, Y ) is obtained as

E( g(X, Y ) ) = sumk suml g(xk, yl)P(X = xk, Y = yl)

58
Q

Linearity of Expectation with multiple variables

A

If X and Y are discrete random variables then

E(X + Y ) = E(X) + E(Y )

59
Q

Independence for Random Variables

A

Two discrete random variables X and Y are independent if
the events “X = xk” and “Y = yl” are independent for all possible values xk, yl

60
Q

Covariance of X, Y

A

The covariance of X and Y is defined by:

61
Q

Correlation coefficient of X and Y

A
62
Q

Formula for Covariance (easy)

A

Cov(X, Y ) = E(XY ) − E(X)E(Y )

63
Q

Normal distribution formula

A
64
Q

Normalisation

A

Using the substitution z = (x − µ)/σ one can confirm that the
p.d.f. is normalised

65
Q

Standardisation

A

When you standardise a normal distribution, the mean becomes 0 and the standard deviation becomes 1

66
Q

z-scores

A

Let Z ∼ N(0, 1) denote a standard normal random variable.

67
Q

Quartiles and the Median

A
68
Q

Z-Score and Standard Normal Distribution Lemma

A
69
Q

Moment Generating Function of a Normal Random Variable

A
70
Q

Properties of Expectation and Variance for Independent Discrete Random Variables

A
71
Q

Variance of a Linear Combination of Independent Discrete Random Variables

A
72
Q

Properties of Correlation for Discrete Random Variables

A
73
Q

Central Limit Theorem

A
74
Q

Sum of Independent Normal Random Variables

A

If X 1 ,X 2 ,…,X n are independent normal random variables with mean μ and variance
σ2, the sum X=X1​+X2+⋯+Xn is also a normal random variable. The distribution of X is given by X∼N(nμ,nσ2), meaning the mean of
X is nμ and the variance of X is 2 nσ2

75
Q

What is a survey?

A

A survey is the collection of data from a sample of the population.

76
Q

What is an observational study?

A

In an observational study researchers observe the behaviour of individuals
without trying to influence the outcome of the study

77
Q

What is an experiment?

A

In a designed experiment researchers apply some treatment to the units under
investigation and measure the response.

78
Q

Quantitative data

A

Continuous variables/data are variables which are given in
terms of real numbers

Discrete variables/data are variables which are given by integers

79
Q

Qualitative data

A

Categorical variables/data are
variables which are expressed in terms of categories

Ordinal variables/data are variables which are expressed in
terms of ordered categories

80
Q

Sample Mean

A

This is called the estimator of the mean where x bar is called the point estimate

81
Q

Median

A

If n is odd Q2 = x (n+1/2)

If n is even Q2 = average (x(n/2),x((n/2) +1)

82
Q

Population Variance

A

This is a biased estimator

83
Q

Sample Variance

A
84
Q

Interquartile Range

A
85
Q

Five-number summary

A

mx, Q1, Q2, Q3, Mx

86
Q

Sample covariance

A
87
Q

Sample linear correlation coefficient

A
88
Q

What is a statistic?

A
89
Q

Expectation and Variance of Sample Mean

A
90
Q

Sample Proportion

A
91
Q

Unbiased Estimator

A

An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter

92
Q

Bias in Estimation of Variance

A

That means Σ2is biased and it slightly underestimates the variance

93
Q

Mean Square error

A

Measures the average squared difference between the estimated values and the actual value

94
Q

Confidence interval for the sample mean

A
95
Q

Level of confidence

A

The fraction of 1−α of such intervals contain the population mean µ. We call 1−α the level of confidence

96
Q

Confidence interval for the sample proportion

A
97
Q

What is a hypothesis?

A

A hypothesis is a statement about a parameter θ of
the pmf (or pdf ) of a random variable X

98
Q

Null Hypothesis

A

In hypothesis testing the central claim, the null hypothesis H0 is a statement θ = θ0 which we intend to find evidence against

99
Q

Significance level

A

Assume the null hypothesis H0 is valid. We say a type-I error has occurred if the test procedure for H0 rejects the null hypothesis. The probability of a type-I error occurring is called the significance level α of the
test procedure

100
Q

Alternative hypothesis

A

The test procedure tests the null hypothesis H0 against the so called alternative hypothesis H1. The alternative hypothesis specifies under which conditions the null hypothesis should be rejected.

101
Q

Two-tailed test

A

If we test H0 : θ = θ0 against H1 : θ /= θ0 we need a two-sided
test.

102
Q

Right-tailed test

A

If we test H0 : θ = θ0 against H1 : θ > θ0

103
Q

Left-tailed test

A

If we test H0 : θ = θ0 against H1 : θ < θ0

104
Q

Type-I error

A

A type-I error occurs if the test rejects the null hypothesis H0 even though H0 is valid

105
Q

Type-II error

A

the test does not reject H0 even though in some sense H0 is false

106
Q

Definition of the Power

A

The probability for a type-II error is denoted
by β. 1−β is called the power of the test

107
Q

Definition of P-value

A

the P-value of the observation is the probability to observe a sample statistic as extreme or more extreme as the observation under the assumption that
the null hypothesis is true

At significance level α the null hypothesis is rejected if the P-value obeys P < α.

108
Q

Interpretation of P-values

A
109
Q

Implications of Conditional Probability

A
110
Q

The multiplication rule

A
111
Q

The partition of events

A
112
Q

Law of total probability

A
113
Q

Total Probability for Conditional Events

A
114
Q

Bayes’ Theorem

A
115
Q

Conditional expectation

A
116
Q

Law of total probability of expectations

A

Mathematics (5 decks)

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