Introduction to Probability and Statistics - Review Sheet | STAT 350, Study notes of Mathematical Statistics

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Complement Rule: For any event A, P (A) = 1 − P (A′). The Impossible Event: If A and B are mutually exclusive, then P (A ∩ B) = 0. Addition Rule: For any two events A and B,

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

Counting Theorems:

nPk =^ (n n−! k)!

nCk =

( n

k

)

= (^) k!(nn −! k)!

Conditional Probability:

P (A|B) = P^ ( PA (^ ∩B^ )B)

Multiplication Rule: P (A ∩ B) = P (A|B)P (B) Law of Total Probability: For a partition A 1 ,... , Ak and event B P (B) =

∑^ k

i=

P (B|Ai)P (Ai)

Bayes’ Theorem: For a partition j = 1,... , k: A 1 ,... , Ak , event B, and

P (Aj |B) = ∑ kP^ (B|Aj^ )P^ (Aj^ )

i=

P (B|Ai)P (Ai)

Multiplication Rule for Independent Events: If A 1 ,... , Ak are mutually independent, then

P (A 1 ∩ · · · ∩ Ak ) =

∏^ k

i=

P (Ai)

indent Cumulative Distribution Function: Discrete:

F (x) = P (X ≤ x) =^ ∑

y;y≤x

p(y) P (a ≤ X ≤ b) = F (b) − F (a−)

Continuous: F (x) = P (X ≤ x) =

∫ x

−∞

f (y) dy P (a ≤ X ≤ b) = F (b) − F (a) (100p)th^ Percentile (Continuous Distribution): For 0 < p < 1, the percen tile x solves the equation F (x) = p Expectation: Discrete:

μ = E[X] =^ ∑

x∈D

x p(x)

E[h(X)] =^ ∑

x∈D

h(x)p(x)

σ^2 = V [X] =^ ∑

x∈D

x^2 p(x) − μ^2 Continuous: μ = E[X] =

∫ ∞

−∞

xf (x) dx

E[h(X)] =

∫ ∞

−∞

h(x)f (x) dx

σ^2 = V [X] =

∫ ∞

−∞^ x

(^2) f (x) dx − μ 2 Binomial: b(x; n, p) =

( n

x

)

px(1 − p)n−x Poisson: p(x; λ) = λx xe!− λ, for x = 0, 1 ,... , ∞ Uniform: f (x) = (^) B −^1 A for A ≤ x ≤ B Exponential: f (x) = λe−λx^ for x > 0 F (x) = 1 − e−λx^ for x > 0

For a joint probability mass functionrandom variables X and Y : p(x, y) of two discrete Marginal probability mass function of X:

pX (x) =^ ∑

y∈D

p(x, y)

Conditional probability mass function of X given Y = y: pX|y (x) = p p(Yx, y (y))

Expectation:

E[h(X, Y )] =^ ∑

x

∑

y

h(x, y)p(x, y)

Covariance:

Cov(X, Y ) =^ ∑

x

∑

y

xy p(x, y) − E[X]E[Y ]

Correlation: ρX,Y = Cov σX(X, Y σY^ )

Distribution of the Sample Mean: Let X 1 ,... , Xn be a random sample from a population with mean σ. μ and standard deviation

Any Population, Any Sample Size:

E[ X¯] = μ, V [ X¯] = σ n^2

Normal Population, Any Sample Size: X¯ is normal with mean μ and variance σ n^2. Any Population, Large Sample Size: X¯ is approximately normal with mean μ and variance σ n^2. (Approximate) Confidence Intervals for μ: Any Population, Large Sample Size:

X¯ ± zα/ 2 √^ Sn

Normal Population, Any Sample Size:

X¯ ± tn− 1 ,α/ 2 √^ Sn

(Approximate) Confidence Interval for a Proportion:

pˆ ± zα/ 2

√

pˆ(1 − pˆ) n

Hypothesis Test for μ (Large Sample): Test Statistic: Z = ¯x^ −√ S^ μ^0 n Hypothesis Test for μ (Normal Population, unknown σ): Test Statistic: T = ¯x^ −√ S^ μ^0 n

Hypothesis Test for p (Large Sample): Test Statistic:

Z = √ pˆ^ −^ p^0

p 0 (1 − p 0 ) n (Approximate) Confidence Interval for μ 1 − μ 2 : ( X¯ 1 − X¯ 2 ) ± E where

(i) E = Zα/ 2

√ σ 2

n^1 1 +^

σ 22 n 2 for^ σ^1 , σ^2 known, Normal Populations, any Sample Sizes (ii) E = Zα/ 2

√ S 2

n^1 1 +^

S 22

n 2 for Large Sample Sizes, any two Populations (iii) E = tα/ 2 Sp

√ 1

n 1 +^

n 2 , S^2 p = (n^1 −^ 1) n 1 S +^21 + (n 2 n−^2 2 − 1)S^22 for unknown σ 1 = σ 2 , Normal Populations, any Sample Sizes (Approximate) Confidence Interval for p 1 − p 2 :

(ˆp 1 − pˆ 2 ) ± zα/ 2

√

pˆ 1 (1 − pˆ 1 ) n 1 +

ˆp 2 (1 − pˆ 2 ) n 2 for Large Sample Sizes (Approximate) Hypothesis Test for μ 1 − μ 2 :

Z = X¯ √^1 − σ^ X 2 ¯^2 −^ ∆^0

n^1 1 +^

σ^22 n 2

; Z = X¯ √^1 − S^ X 2 ¯^2 −^ ∆^0

n^1 1 +^

S^22

n 2

;

T = X¯^1 −^ X¯^2 −^ ∆^0

Sp

√ 1

n 1 +^

n 2 (Approximate) Hypothesis Test for p 1 − p 2 :

Z = √ pˆ^1 −^ ˆp^2

p ˆ(1 − ˆp)( n^11 + (^) n^12 )

, pˆ = n^1 n^ pˆ^11 ++^ nn^22 pˆ^2