Intro to Probability and Statistics Formula Sheet

Chapter 2

Union

= P(A) + P(B) + P(C) – P(A∩B) – P(A∩C) – P(B∩C) + P(A∩B∩C)

Intersection:

(3) can also sum up 2 random variables

Derived from:

NOTE: if we can already find P(A ∩ B) without using Bayes’ Theorem, then do it

Bayes Theorem:

De Morgan’s Law

(A U B)’ = A’ ∩ B’ (A ∩ B)’ = A’ U B’

Chapter 3 – Discrete RV and Distributions

( ) ( ) ( )

Discrete Uniform Distribution

Binomial Distribution

Geometric Distribution

Memoryless property:

P(Z > s + t | Z > s) = P(Z > t)

Poisson Approx. to the Binomial Distribution (use when p is small)

NOTE: λ = np

For cases (Poisson):

P(X ≥ x) 1 – Σ f(x) e.g. P(X ≥ 2) 1 – f(1) - f(2)

P(X ≤ x) Σ f(x) e.g. P(X ≥ 2) f(1) + f(2)

VERY IMPORTANT: **when question asks us to “estimate” use either Poisson or Normal approximation to the binomial distribution! (for small p use Poisson, for large p use Normal)**

Chapter 4 – Continuous RV and Distributions

Cases:

P(X < x) = ∫ ( )

for X > Y

e.g. P(X < 2) = ∫ ( )

for X > 1

P(X > x) = 1 ∫ ( )

for x > y OR ∫ ( )

for X > Y

e.g. P(X > 2) = 1 – ∫ ( )

for X > 1 OR ∫ ( )

for X > 1

For cases that we have Cumulative Distribution Function of value X > y, use the former

The Cumulative Distribution Function is the integral of the Probability Density Function – we can use this to solve problems too (e.g. for P(X < y) = F(y); for P(X > y) = 1 – F(y)

When we have the Cumulative Distribution Function, F(x) = 1 at value x > y, when we compute any F(X > y) then F(x) = 0

Continuous Uniform Distribution

Cumulative Distributive Function of a Continuous Uniform Distribution

NOTE: the middle formula is the same as taking the integral of f(x)

Normal Distribution

Normal Apprx. to Binomial Distribution (use when p is large)

( ≤ ) ( .

√ (1 ) ≤

+ .

√ (1 ))

NOTE: when we have cases like P(X = 10) = P(9.5 < Z < 10.5)

also when we have the question “when A is more than 500” we compute X ≥ 501, meaning we will get P(Z ≥ 501 – 0.5 + np/sqrt(np(1-p))

Exponential Distribution

Cumulative Distributive Function of an Exponential Distribution

**remember that MEAN = 1/λ

also P(X<15 | P(X>10) = P(10<X<15)/P(X>10) refer to conditional probability.

Cases: P(Z < x) = Φ(x) P(Z > x) = P( Z < -x) = Φ(-x) = 1 – Φ(x); P(y < Z < x) = Φ(x) – Φ(y) OR P(y < Z < x) = Φ(x) – (1 – Φ(-y))

Chapter 5 – Joint Distribution Discrete Case

Continuous Case

NOTE: Don’t forget to use the conditional probability distribution indicated above

Independence

More than two RVs

Covariance/Expected Values of 2+ Variables

That is, E[h(X, Y)] can be thought of as the weighted average of h(x, y) for each point in the range of (X,Y). The value of E[h(X,Y )] represents the average value of h(X,Y ) that is expected in a long sequence of repeated trials of the random

experiment.

VERY IMPORTANT: **for limits of integration for expected value we use the whole range’s not just either x or y**

Transforming Random Variables

Discrete Case

Continuous Case

1. find y = h(x) = the equation we want to convert x to such as y = x2 2. find the inverse of h(x), h-1(x) = u(y) = x 3. find the Jacobian, which is u’(y)

note that by definition J = fY(x)p 4. then fY(y) = fX[u(y)] |J|