QTM: Probability Distribution and Normal Distribution

Should it surprise you if the actual outcome of a potential random variable problem is slightly below or under the probability you calculated?

No— there is natural variability

Random variable

Random process with numerical outcome

Expected value

Sum of X*P(X)

Expected value can also be known as

Mean

How to find proportion of certain bin in continuous distribution (histogram)

See count divide by total sample size

Probability density function

Smooth curve over bars on histogram

total area under a curve in density distribution

1

Two types of random variables

Discrete and continuous

Probability distribution in the real world shows you

Your chances of winning

Columns for probability distribution

x, p(x), x*p(x), x²*p(x)

When you have money you should always round

Up

Variability

Difference in value of random variable (how much variation is expected)

Why do you need to find variability?

To find standard deviation

Formula for variability (to get ______)

SD*2= sum of x²*p(x)-(x*p(x))²

Fair game

costs as much as payout so expected profit =0

How do you prove a game is fair?

Sum of X*p(x) =0

Normal distribution

Most variable are normally distributed aka uni-modal and bell shaped

Two ways to prove symmetry

1. Mean= median=mode

2. Peason’s index

Pearson’s Index

I=[3(mean-median)]/SD

*must be between -1 and 1

What does z score tell you

How far away you are from the mean

Why do you need z score?

Standardize data to compare

Percentile

Percentage of data below specific data point

How do you round percentile?

Always round up

Gold standard for z-score

Mean =0

SD = 1

Z score formula

Z= (given-average)/SD

How to use z-score

Find z score then look at chart for probability and percentile

unusual vs outlier stats wth z score

|z|>2 = unusual

|z|>3= outlier

Empirical Rule

68-95-99.7

1 SD-2 SD- 3 SD

Area under curve

Probability

How do you get right side of a Norma distribution

Subtract from 1

How to look at normal distribution graph

Left to right (DO NOT START FROM MEAN)

How do you find cutoff?

1. Give you probability => look for closest z score (NUMBER IN PROBLEM IS NOT Z SCORE)

2. Plug into equation (x or given should be blank)

3. Solve algebra

Upper vs lower limit questions normal distribution

Will be 2 SD away BECAUSE otherwise they are outliers and we do not plot outlier (only use 3 SD if it says to include outliers)

How to write up normal distribution with parameters

N (mean =0, SD =1)

How to compare z scores

Higher |z| score = unusual