ECE 302 Exam 2 Study Guide

How do you find the joint cdf given the joint pdf?

(integral from -infinity to y) (integral from -infinity to x) fR,S(r,s) dr, ds Just make sure the -infinity to x matches the variable substitution you did for x and same for -infinity to y

What is Var(X)? (variance)

E(X - µ)^2 where µ is the mean of the random variable. Can also be E[X^2] - E[X]^2

What is E[X + a]?

E[X + a] = E[X] + a where "a" is a constant

What is the formula for expectation in continuous?

E[X] = integral over all values (x * fx(x) )

What is the formula for expectation in discrete?

E[X] = summation x * p(x)

What is E[aX + b]?

E[aX + b] = aE[X] + b where "a" and "b" are constants

What is E[aX]?

E[aX] = a * E[X] where "a" is a constant

What is the P(a<= x <= b)?

Fx(b) - Fx(a) where Fx(X) is the CDF

Do you care about order for combinations?

No. Order doesn't matter for combinations. It matters for permutations.

What is P(X <= b)?

P(X <= b) = Fx(b)

How do you find the probability given the joint pdf?

Probability of A = double integral over the Area (A) of fX,Y(x,y) dx, dy

If given a cdf, how do you find the pdf?

Take the derivative of the cdf to get the pdf

If given a pdf, how do you find the cdf?

Take the integral of the pdf to find the cdf

What is Var(X + a)? (variance)

Var[X] Adding a constant doesn't do anything to variance because variance is essentially just like standard deviation. It's like if you add 10 to everyone's exam scores, then that doesn't change the spread of the exam grades i.e. the standard deviation (variance).

If you have to find the constant "c" within a given joint pdf (such as fX,Y(x,y) = c *x * y), how do you figure out what "c" is?

You use the fact that the double integral from -infinity to infinity (for both integrals) of fX,Y(x,y) = 1. So you just get the double integral over -infinity to infinity for both dx and dy and then solve for the constant "c".

What is Var(cX)? (variance)

c^2 * Var(X)

If 2 random variables (X and Y) are independent, then what is fX,Y(x,y) equal to?

fX,Y(x,y) = fX(x) * fY(y) when X and Y are independent variables

How do you find the pdf of Y (fY(y)) ?

fY(y) = (d/dy) * FY(y) = (d/dy) * FX(y) = FX(y)' (prime) * (d/dy) (y) = fx(y) * (d/dy) * y

How do you find the marginal pdf given the joint pdf?

fY(y) = integral over all x (fX,Y(x,y) ) dx fX(x) = integral over all y (fX,Y(x,y) ) dy

What is E(g(X))?

integral over all values (g(x) * fx(x) )

How do you find the marginal CDF of y?

lim (x -> infinity) of FX,Y(x,y) = FY(y)

How do you find the marginal CDF of x?

lim (y -> infinity) of FX,Y(x,y) = FX(x)

What does n and k stand for in nCk?

n is the total pool of things to choose from and k is the number of things you choose

What is the formula for combination?

n!/k!(n-k)! for nCk where n is the total pool and k is the number of things you choose

If 2 random variables (X and Y) are independent, then what is pX,Y(x,y) equal to?

pX,Y(x,y) = pX(x) * pY(y) when X and Y are independent variables

What is pmf?

probability mass function

What is the pmf of a Bernoulli Distribution?

px(0) = 1-p and px(1) = p where p is the probability of success

What is the pmf of a Binomial Distribution?

px(k) = nCk * (p^k) * (1-p)^(n-k) where n is the total pool of things to choose from and k is the number of things you choose and p is the probability of success in the (p^k) and (1-p) terms

what does Fx(X) represent?

the cdf

what does fx(X) represent?

the pdf

what is px(0) for Bernoulli Distribution?

the pmf px(0) = 1 - p

what is px(1) for Bernoulli Distribution?

the pmf px(1) = p