Chapter 5 (Continuous Probability Distributions)

Question 1

What are some facts about the Normal Distribution?

Answer 1

• it’s continuous between -infinity and + infinity
• there are two parameters: myu and the variance (sigma^2)
• curve is bell shaped
• symmetric around myu
• it’s very common
• pdf is not an integral by hand - use a table to find areas under the curve.

Question 2

Why is the normal distribution so common?

Answer 2

1) lots of populations have normal distributions.
2) sims of means of random variables have normal distributions under certain conditions regardless of the underlying population distribution.

Question 3

How do we denote the Normal Distribution?

Answer 3

denoted: X~N(mu, sigma^2)

Question 4

What is the normal curve rules of thumb?

Answer 4

68 - 95 - 99.7 Rule!

68% lie within 1 SD
95% lie within 2 SDs
99.7% lie within 3 SD/

Question 5

How do you find a “z-score”?

Answer 5

X-muu
–––––––
sigma

Or
Value - mean / standard deviation

Question 6

How to notate P(Z ≤ z) with phi?

Answer 6

Phi(z)

Question 7

Sums of independent, normally distributed random variables are…

Answer 7

Also normally distributed.

Question 8

If many random variables Xi are independent, then Xbar has what distribution?

Answer 8

Xbar ~ N(mu, sigma^2/n)

Question 9

What is the central limit theorem?

Answer 9

If X_1, X_2, …, X_n, is a sample (ind. random variables from one population) from some distribution with mean muy and variance sigma^2, if the sample size n is large enough, the distribution of their average is approximately normal:

X_bar~N(myu, sigma^2/n)

Question 10

What does “iid” mean?

Answer 10

Independent and identically distributed (cane from same population (distribution)).

Question 11

What is the distribution of mean (X_bar) according to the central limit theorem?

Answer 11

X_bar~•N(mu, sigma^2/n)

~• is approximately distributed

Question 12

What is the sum (nX_bar) according to the central limit theorem?

Answer 12

nX_bar ~• N(n * mu, n*sigma^2)

Question 13

What is the proportion (p_hat) according to the central limit theorem?

Answer 13

P_hat ~• N(p, p*(1-p)/n)

Question 14

What are the conditions for the CLT?

Answer 14

1 and 2) the random variables are iid. (Independent and identically (same)distributed)

3) the sample size n is large enough (usually 25+)
4) Were concerned with the mean (X_bar), sum (n*X_bar), or proportion (p_hat)

Question 15

Under certain circumstances a Binomial distribution may look like a normal distribution. What are these circumstances?

Answer 15

X•~ N(np, np(1-p))

Where np ≥ 5
and np(1-p) ≥ 5

Question 16

If Z = (X-mu)/sigma, what distribution does Z have?

Answer 16

A normal distribution!

Z~N(0,1)