chapt functions of random variables

aaaaaaaaa i dont want to do this omg

what is it considered if Y = g(X) and is a single variable and what does it mean

g is monotonic increasing

X is known exactly and that would mean Y is also known exactly

how do you find the probability of the PDF of X and the PDF of Y from the graph of a monotonically increasing function

probability of f_X(x) = f_X(x)dx

probability of f_Y(y) = f_Y(y)dy

give the PDF of Y in terms of the PDF of X

f_Y(y) = f_X(x) |dx/dy| , where x = g^-1(y)

how do you find dx/dy

1. differentiate y with respect to x 

2. take the reciprocal of the differential of y with respect to x

what will be the result when there is multi-valued single variable function

the probability of f_Y(y)dy is mapped to 2 regions → f_X(x₁)dx₁ and f_X(x₂)dx₂

what is the PDF of y of a multi-valued single variable function

f_Y(y) = f_X(x₁) |dx₁/dy| + f_X(x₂) |dx₂/dy|

what is the formula for multi-valued functions of the pdf of Y

f_Y(y) = Σ f_X(x_i) |dx_i/dy|

what is the result of the mean and the standard deviation when a random variable is scaled eg Y = bX

μ_Y = bμ_X

σY = |b|σ_X

what is the formula for covariance of the 2 random variables 

cov (X, Y) = E[(X - μ_X)(Y - μ_Y)]

what is the correlation coefficient formula from covariance

ρ_XY = cov (X, Y) / σ_X σ_Y

what is the range of ρ

-1 ≤ ρ ≤ 1

what is the relationship between correlation and dependence

being independent → uncorrelated 

but uncorrelated does not mean that it is independent 

what is the new mean of a normal distribution Y, where 2 normal distributions, a₁X₁ and a₂X₂ add up

μ_Y = a₁μ_X1 + a₂μ_X2

what is the new variance of a normal distribution Y, where 2 normal distributions a₁X₁ and a₂X₂ add up

σ²_Y = a₁²σ²_X1 + a₂²σ²_X2 + 2 a₁a₂ ρ_X1X2 σ_X1σ_X2