Module 13: Normal Random Variables

Question 1

`summarize a normal random variable

Answer 1

Scenario it models: siutations where data is symmetrically distributed about amean value, where the closer an observation is the to the mean, the greater the likelihood of it occuring

PDF: f(x) = 1/(sqrt(2pi) sigma) * e^(-((x-mew)^2)/(2(sigma^2)))

E(X) = mew
Var(X) = sigma ^ 2

Question 2

does the standard normal distribution look like and how to do interpret it?

Answer 2

the normal curve. 68% of the area is 1 s.d. away from the mean 34.13% each direction, when 95% for 2 s.d’s 13.59% each + the 34.13% then 99.7 is 3 s.d. away with 2.14% each + the other 2 then 100% 4 s.d. away 0.0014% away plus the other 3.

Question 3

what is the thrm for normality being preseved by linear transformations?

Answer 3

Let X be a normal RV with mean mew and variance sigma^2, meaning its PDF is: f(x) = 1/(sqrt(2pi) sigma) * e^(-((x-mew)^2)/(2(sigma^2)))

Then for any constants a ≠ 0 and b, the random variable Y = aX + b is also normal with E(Y) = aμ + b and Var(Y) = a^2 * σ^2

Question 4

what is the corollary for the z of the standard normal random variable?

Answer 4

If X is a normal random variable with mean μ and variance σ^2, the random variable Z defined as: Z = (X - μ)/σ, is also a normal with a mean of 0 and varaince of 1. Z is called the standard normal random variable.