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Summary Introduction to Probability and Statistics

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Institution
Simon Fraser University (SFU )

I have summarized some of the corresponding books along with my interpretation of the lecture notes from the course. This is suitable for anyone who is new to the idea of probability and statistics. It assumed knowledge of derivatives and integrals.

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Summarized whole book? No
Which chapters are summarized? 1-9
Uploaded on January 3, 2022
Number of pages 4
Written in 2021/2022
Type Summary

probability
statistics
discrete
continuous
joint
pmf
pdf
marginal pmf
marginal pdf
cdf
joint setting
confidence intervals
sample
population
hypothesis testing
sample size calculations
bernoulli
random

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1 Introduction
Population - The de
ned set of object we desire information from. We can have a population mean µ, median µ̃,
variance σ2 , and standard deviation σ.
Sample - The subset of the population we analyze. We can have a sample
n
mean (arithmetic average of
P
x i
x̄ = n
i=1

n 2
data), median x̃ (the middle value of an ordered data set), variance s2 = with standard deviation s. The
P
i=1 (xi −x̄)
√ n−1
standard error of the mean is s/ n.
Variables are either quantitative, continuous or discrete, or qualitative, categorical (ordinal or nominal) or unstruc-
tured.
A multimodal histogram has multiple peaks. A dichotomous population consists of only two categories; the
sample proportion in category one is x/n, and 1 − x/n in category two.
Counting
Order with replacement: nk . Order without replacement: (n−k)!
n!
. Order with non-distinct objects of K sets: n!
n1 !...nK ! .
No order with replacement: n+k−1
. No order without replacement: nk .

k

2 Probability
An event is a subset of outcomes contained in the sample space which is the set of all possible outcomes.
Union A ∪ B
Intersection A ∩ B
Disjoint Events - when A ∩ B = ∅
Probability of A from the sample: P (A) = cardinality
cardinality S .
A

DeMorgan's Laws (A ∪ B)′ = A′ ∩ B ′ and (A ∩ B)′ P = A′ ∪ B ′ .
Axioms P (A) ≥ 0, P (S) = 1, and P (A1 ∪ A2 ∪ ...) = ∞ i=1 P (Ai ) for a collection of disjoint events.
For any event A, P (A) + P (A′ ) = 1 and P (A) ≤ 1.
Union of Three Events - P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (A, B) + P (A, C) + P (B, C) + P (A, B, C).
Conditional Probability - P (A|B) = (A,B)P (B)
Independence - A and B are independent events if: P (A, B) = P (A)P (B), P (A|B) = P (A), or P (B|A) = P (B),
mutual independence extends for k events..
Intersection via Conditioning (Multiplication Rule) - P (A, B) = P (B|A)P (A)
Pk
LoTP - For any set of disjoint and exhaustive events A1 , ..., Ak , P (B) = P (B|A1 )P (A1 ) + · · · + P (B|Ak )P (Ak ) =
i=1 P (B|Ai )P (Ai )
Bayes' Theorem - Let A1 , ..., Ak be any set of disjoint and exhaustive events with prior probabilities P (Ai ). The
posterior probability of Aj given B is P (Aj |B) = P (Aj ,B)
P (B) = Pk
P (B|Aj )P (Aj )
i=1 P (B|Ai )P (Ai )
Random Variables (RV's) and Distributions
RV - A function whose domain is S and whose range is R.
Realization - For some outcome ω and rv X , X(ω) = x is the realization.
PMF - Gives theP probability that a discrete rv takes on x, de
ned as P (X = x) = fX (x). A PMF satis
es
fX (x) ≥ 0 ∀ x and fX (x) = 1.
x
PDF - A function f( x) for a continuous rv such that P (a ≤ X ≤ b) = a f (x)dx. A PDF satis
es f (x) ≥ 0 ∀ x and
Rb

f (x)dx = 1. If the CDF is dierentiable, then the PDF is the derivative of the CDF.
R∞
−∞
Expected Values (of a function of a rv) - (X = x) for discrete X and E[g(X)] =
P
E[g(X)] = x g(x)P
g(x)f (x)dx for continuous X .
R∞
−∞
Linearity of EV's - E[aX + bY + c] = aE[X] + bE[Y ] + c
Variance - V ar[X] = E[X − E[X]]p2 = E[X 2 ] − (E[X])2
Standard Deviation - SD[X] = V ar[X] P
Discrete CDF - FX (x) = P (X ≤ x) = y:y≤x fX (y), the observed value of X will be at most x. Note that
P (a ≤ X ≤ b) = F (b) − F (a−), where F (a−) includes P (X = a) for discrete X .
Continuous CDF - F (x) = P (X ≤ x) = −∞ f (y)dy (integrate the PDF). Note that P (X > a) = 1 − F (a) and
Rx

P (a ≤ X ≤ b) = F (b) − F (a).
Properties of CDFs - limx→−∞ P (X ≤ x) = 0, limx→∞ P (X ≤ x) = 1, P (X ≤ x) is non-decreasing, and P (X ≤ x)
is right continuous.

1

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