chapt random variables

broooo what the helly

what are the 2 types of random variables

discrete and continuous

what is the distribution function for a probability distribution of a random variable

cumulative distribution function (CDF)

what is the characteristics of a cumulative distribution function

continuous and non-decreasing with x

what is the probability distribution for a discrete random variable 

probability mass function (pmf)

what is the relationship between the CDF and the pmf

CDF = F_x (xn) = Σ p_x(xn) 

what is the probability distribution of a continuous random variable

probability density function

what is the characteristics of a probability density function

• cannot be negative

• area under the curve = 1

• units of 1/X eg if X~ m, f_X(x) ~ m^-1

• value represents gradient of CDF

what is the characteristics of the cumulative distribution function

• ranges from 0 to 1 (F_X(-∞) = 0, F_X(+∞) = 1)

• montonically increasing

• dimensionless

• gradient given by pdf

what is the area under the f(x) (pdf) curve

always 1

how do you find the CDF from the pdf equation

integrate the pdf with lower limit and variable to find and upper limit

what is the point at which x the PDF is the maximum

mode

what is the point at which the data is divided into half 

median

what is some weighted value of the spread of the data

the mean value

what is the formula for mean / expected value of X

E[X] = Σ [x_i]p_X(x)]

E[X] = ∫ x f_x (x) dx

the sum of all the probabilities and the frequency of the probability that it happened

what is the expected value of a function g(x)

E[g(x)] = Σ [g(x)] p_X(x)

E[g(x)] = ∫g(x) f_x (x) dx

multiply the function with the probability 

what is the formula for variance σ²

(also known as the second central moment of X)

E[(X - μ_X)²] = E[X²] - μ_X²

∫ (x - μ_x)² f_x(x) dx

Σ (x - μ_x)² p_X(x)

what is the coefficient of variation δ_x

δ_x = σ_x / μ_x

what is the third central moment of X formula

E[(X - μ_X)³] 

Σ (x - μ_x)³ p_X(x)

∫ (x - μ_x)³ f_x(x) dx

what is the formula for skewness θ

E[(X - μ_X)³] / σ_X³

what does it mean about the skewness when the distribution is symmetric

symmetric distribution implies skewness = 0

skewness = 0 does not necessarily imply symmetry