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Mathe > Probability & Statistics > Flashcards

Probability & Statistics Flashcards

1
Q

What is one-way data?

A

Data that’s focused on or collected about just the single individual.

Variable data given for individuals.

One independent variable, called individuals, and one OR more dependent variables, called the variables

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2
Q

What does this graph represent?

A

Bar Chart

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3
Q

What does this graph represent?

A

Histogram

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4
Q

What does this graph represent?

A

Line graph

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5
Q

What does this graph represent?

A

Ogive

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6
Q

What is Two-way data?

A

Two independent categories on which the variables are dependent.

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7
Q

What is the relative frequency table?

A

A table that shows percentages instead of actual counts of outcomes of one experiment.

Displays data in Two-way or One-way tables as percentages.

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8
Q

What is Joint Distribution Table?

A

A table that compares two different distributions.

It helps us see a correlation between the two variables (distributions).

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9
Q

What is Marginal Distribution?

A

The Total row or the Total column in a Joint Distribution.

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10
Q

What is Conditional Distribution?

A

Distribution of one variable, given a particular value of the other variable.

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11
Q

What is the joint distribution section of this table?

A

The joint distribution or the joint probability distribution is the probability that a pair of events can happen. All of the possible pairs of events happen in the body of the table

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12
Q

What is the marginal distribution section of this table?

A

The marginal distribution comes from the total column OR total row.

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13
Q

What is the Frequency Table?

A

A table that displays how frequently or infrequently something appears.

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14
Q

What is a dot plot?

A

Shows frequency of small datasets.

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15
Q

What is a Histogram

(or Frequency Histogram) ?

A

Just like a Bar Graph, except that we collect the data into buckets or bins, and then sketch a bar for each bucket.

Representation of the distribution of numerical or categorical data in bins.

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16
Q

What is a Stem Plot?

A

Data grouped together by the first digit(s) in each number.

The “stems” are the numbers on the left, in this case, the 6 and the 7.

The “leaves” are all the other numbers on the right.

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17
Q

What is Mean?

A

The average.

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18
Q

What is median?

A

The value in the middle when you line up all the data in order.

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19
Q

What is Mode?

A

The value that appears most often.

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20
Q

What do we need to look at first when we analyze data?

A
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21
Q

What is spread?

A

How and by how much a data set is spread out around its center.

We call measures of spread measures of dispersion or scatter.

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22
Q

What is Range?

A

The difference between the largest and smallest value.

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23
Q

What is Quartile?

A
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24
Q

What is Interquartile range?

A

The difference between the median of the upper half

and the median of the lower half.

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25
Q

What happens to measures of central tendency and spread when we add/remove a constant value to every value in the data set?

A

Adding 6 to the entire data set also adds 6 to the mean, median, and mode, but the range and IQR stay the same.

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26
Q

What happens when we multiply our data set by a constant value?

A
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27
Q

What happens with the mean if we add a data point to our data set?

A
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28
Q

What do Outliers do to the mean and median of a data set?

A

Change Mean significantly, change Median slightly

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29
Q

Explain this plot.

A
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30
Q

What is population and sample?

A

Population is the entire group or all of the subjects in a population.

Sample is a subset of the population and we hope that it will be somewhat representative of the population as a whole.

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31
Q

Describe Mean, Variance, Standard Deviation,

biased/unbiased Sample Standard Deviation.

A
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32
Q

What is Frequency Polygon?

A
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33
Q

What is Density Curve?

A

Shows the distribution of the data when we increase the number of bins in a histogram to infinity.

Visualization of a distribution where the data points can take on any value in a continuum.

The area under the density curve is equal to 100 percent of all probabilities.

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34
Q

Draw a symmetric distribution.

A
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35
Q

Draw a Left Skewed Distribution

A
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36
Q

Draw a Right Skewed Distribution.

A
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37
Q

What should be used for measuring central tendency and spread if we have skewed data?

A
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38
Q

What is 1.5 - IQR Rule?

A
39
Q

Describe Normal Distribution, Percentile, z-Score, z-Score tables (+/-)

A
40
Q

Define simple probability of an “event”

A
41
Q

What is the Probability Addition Rule (Sum Rule) and what does it make sure?

A

For mutually exclusive events the probability of

P(AuB) = P(A) + P(B)

For overlapping events, we don’t double-count the overlap.

P(AuB) = P(A)+P(B)-P(AandB)

Definition: Suppose some event E can occur in m ways and a second event F can occur in n ways, and suppose both events cannot occur simultaneously. Then E or F can occur in m + n ways.

42
Q

What are Mutually exclusive (or disjoint) events?

A
43
Q
A
44
Q

Apply Probability Multiplication Rule to dependent events.

Give an example of a Dependent Event

A
45
Q

Give an example of an Independent event

A
46
Q

What is the Multiplication Rule?

A

If events A and B are mutually exclusive (independent),

the probability of event A happening and Event B happening on two separate trials is
P(A and B) = P(A) * P(B)

It’s giving us the probability that multiple independent events happen on consecutive trials.

For dependent events:

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

Product Rule Principle: Suppose an event E can occur in m ways and, independent of this event, an event F can occur in n ways. Then combinations of events E and F can occur in m*n ways.

47
Q

What is the formula for Bayes’ Theorem?

A
48
Q

Use Tree Diagramm method for Bayes’ Theorem to calculate the probability that biased dice was chosen given 6 is already rolled. The biased die has a 50% chance of rolling a 6.

A
49
Q

When should you use the Bayes’ Theorem?

A

When you have P(A|B) but want to find P(B|A)

50
Q

Give formula for Conditional Probability.

A
51
Q

The world around us is full of phenomena we perceive as random or unpredictable.
How do we model these phenomena? (How do we call them?)

A

Outcomes of some experiment.

52
Q

What is Sample Space?

A

Sample Space = all possible outcomes

Sample space is set whose elements describe all outcomes of the
experiment in which we are interested.

The outcomes of some experiments are elements of a sample space Ω.

Sample space of the experiment of tossing of a coin is

Ω = {H, T }

53
Q

What are Events?

A

Events are subsets of Sample Space Ω.
The events will be assigned a probability, a number between 0 and 1 that expresses how likely the event is to occur.

54
Q

What is permutation?

A

The order in which n different objects can be
placed. This is called a permutation of the n objects.

There are
n · (n - 1) · · · · 3 · 2 · 1 = n!
possible permutations of n objects (taken all at the time).

55
Q

What are disjoint events?

A

We call events A and B disjoint or mutually exclusive if A and B have no
outcomes in common; in set terminology: A∩B = ∅. For example, the event L
“the birthday falls in a long month” and the event {Feb} are disjoint.

56
Q

What are Intersection, Union, Complement?

A

The set L∩R is called the intersection of L and R and occurs if both L and R
occur. Similarly, we have the union A∪B of two sets A and B, which occurs if
at least one of the events A and B occurs. Another common operation is taking
complements. The event Ac = {ω ∈ Ω : ω ∉ A} is called the complement of A;
it occurs if and only if A does not occur. The complement of Ω is denoted
∅, the empty set, which represents the impossible event.

57
Q

What does it mean event A implies event B?

A

we say that event A implies event B if the outcomes of A also lie
in B. In set notation: A ⊂ B

A is a subset of B

58
Q

What is probability function?

A

We want to express how likely it is that an event occurs. To do this we will
assign a probability to each event.
Since each event has to be assigned a probability, we speak of a probability function. It has to satisfy two basic properties.

59
Q

What is the probability of a union?

A

For any two events A and B we have
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

60
Q

What is probability of complement of event?

A
61
Q

Suppose we throw a coin two times.

What is the sample space associated with this
experiment?

A

Ω = {H, T } × {H, T } = {(H, H), (H, T ), (T, H), (T, T )}

62
Q

If we consider two experiments with sample spaces
Ω1 and Ω2, what is the sample space of the combined experiment?

A

The sample space of the combined experiment is the set
Ω = Ω1 × Ω2 = {(ω1, ω2) : ω1 ∈ Ω1, ω2 ∈ Ω2}
If Ω1 has r elements and Ω2 has s elements, then Ω1 × Ω2 has rs elements.

63
Q

When we perform an experiment n times, what is the corresponding
sample space?

A
64
Q

What are DeMorgan’s Laws?

A
65
Q

What is the probability that neither E nor
F occurs?

A
66
Q

What is the event “at least one of E and F occurs”?

A

the event E U F

67
Q

What event is neither event 𝐴 nor event 𝐵?

A
68
Q

What are the Distributive Laws?

A
69
Q

What are the Associative Laws?

A
70
Q

X \ Y = ?

(in terms of Y’)

A

X \ Y = X ∩ Yc

71
Q

Laws of the Algebra of Sets

A

Idempotent Laws

A ∪ A = A

A ∩ A = A

Associative Laws

(A ∪ B) ∪ C = A ∪ (B ∪ C)

(A ∩ B) ∩ C = A ∩ (B ∩ C)

Commutative Laws

A ∪ B = B ∪ A

A ∩ B = B ∩ A

Distributive Laws

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Identity Laws

A ∪ ∅ = A A ∪ U = U

A ∩ ∅ = ∅ A ∩ U = A

Involution Laws

(Ac)c = A

Complement Laws

A ∪ Ac = U A ∩ Ac = ∅

Uc = ∅ ∅c = U

DeMorgan’s Laws

(A ∪ B)c = Ac ∩ Bc

(A ∩ B)c = Ac ∪ Bc

Others

A \ B = A ∩ Bc

A = (Bc ∩ A) ∪ (B ∩ A)

A ∩ Bc = A ∪ B \ B

Additivity

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

P(A) = P(A ∩ B) + P(A ∩ Bc) = P(B) + P(A \ B)

72
Q

If the two events A and B are independent, what is P(A∩B) ?

A

P(A∩B) = P(A) * P(B)

73
Q

If the two events A and B are overlapping, what is P(A∩B) ?

A

P(A∩B) = P(A|B) * P(B)

74
Q

Give the formula for binomial coefficients. Calculate an example.

A

Note that (n Chose r) has exactly r factors in both the numerator and the denominator.

75
Q

Give the Lemma 1 for binomial coefficients.

A
76
Q

What is the formula for Permutation of n objects taken r at the time?

A

Permutation(n,r) = n! / (n-r)!

77
Q

Give the formula for Permutations with Repetitions.

A

Permutation(n; n1, n2, …, nr)

We are looking for the permutation of n objects of which n1 are alike, n2 are alike, …, nr are alike.

Permutation(n; n1, n2, …, nr) = n! / n1! n2! … nr!

78
Q

Suppose that we are given n distinct objects and wish to arrange r of these objects in a line.
what is the number of ordered samples with Replacement of size r?

A

r

|—————–|

n * n * n … n = nr

79
Q

What is the number of ordered samples without Replacement of size r?

A

nPr = n! / (n-r)!

80
Q

What is a Combination? Give an example.

A

Combination of n objects taken r at a time is any selection of r of the objects where order doesn’t count.

An r combination of a set of n objects is any subset of r elements.

This is denoted by

C(n,r) or nCr

Example: given set {a,b,c,d}

C(4,3) = 4!/(3!*1!)

abc, abd, acd, bcd

81
Q
A
82
Q

How do we calculate circular permutation?

A

(n-1)!

83
Q

What is the expected value (mean) of a discrete random variable?

A
84
Q

What is the variance of a discrete random variable?

A
85
Q

What are Mean and Std.Dev. for combinations of random variables?

A
86
Q

What is the formula for Probability of Binomial Random Variable?

A
87
Q

What is the classical (A Priori) definition for the probability p of an event E?

A

Then p(E) = n(E) / n(S)

88
Q

What is frequency (A Posteriori) definition for probabilty of an event?

A

Suppose after n repetitions, where n is very large,

an event E occurs m times.

Then p = m / n

89
Q

What is Finite Equiprobable Space?

A

A probability space S where each point

is assigned the same probability.

P(A) = n(A) / n(S)

90
Q

What is Finite Probability Space?

A

Let S be a finite sample space S = {a1, a2, …, an}.

A finite probability space is obtained by assigning to each point ai in S a real number pi called the probability of ai satisfying the following properties:

(i) each pi is nonnegative, that is pi >= 0
(ii) the sum of the pi is 1

The probability P(A) of an event A is defined as the sum of the probabilities of the points in A:

P(A) = Sum(P(ai)

91
Q

What is the formula for odds that something happens?

A
92
Q

What is a finite stochastic process?

A

A finite stochastic process is a finite sequence of experiments where each experiment has a finite number of outcomes with given probabilities.

93
Q

What is Multiplication Theorem for Conditional Probability?

Show an example using a tree diagram.

A
94
Q
A

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