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Quantitative Methods Final Exam Cheat Sheet by

Statistics & Probab­ility Notes

 

Standard Deviation

Quartiles

IQR = Q3 = Q1
Outliers are beyond: Q1 - 1.5 * IQR, Q3 + 1.5 + IQR

Calc Entry for Basic Stats

1-VAR Stats

Correl­ation (R^2 near 1 is better fit)

>CO­MBI­NAT­ORIAL PROBAB­ILI­TY<

4-Digit PIN with repetition = 10^4
4-Digit PIN without repetition = 10!/6! = 5040

Permut­ations

How many ways can 6 people be ranked? Ranking n objects leads to n!
How many ways can 6 people be ranked into 3 places:
n(n-1­)...*­(n-r+1) = n!/(n-r)! or 6 nPr 3

Combin­ations

Combin­ations = choosing a certain number of objects from a given set (no order).
N choose R or N!/(N-­R)!R! or nCr

Probab­ility

Example

3 cities from 15 are chosen randomly for a visit. A: 4 cost $800, B: 5 cost $300, C: 6 cost $100. What is the probab­ility that the tour will cost $1000 or less?
- All from C 6 nCr 3
- Two from C, one from B or A (6nCr2­)(5­nCr1) + (6nCr2­)(4­nCr1)
- One from C, two from B (6nCr1­)*(­(5nCr2)
- None from C, three from B (5nCr3)
Compute and sum: 20+75+­60+­60+­30=245; Divide by the total 15nCr3 = 455
.538 or 53.8%

>RANDOM VARIAB­LES<

Example of Random Variable
X is the number of heads in 10 flips of a coin. P(X=4)?
(10nCr­4)/2^10 = .205
X is sum of two dice.
P(X=5) = 4/36
P(9<=X­<=11) = P(X=9)­+P(­X=1­0)+­P(X­=11)= 4/36+3­/36­+2/36

Binomial Random Variables

Binomial Random Variable represents the number of successes in n trials with probab­ility p of success. The probab­ility of 4 successes in 10 trials (p=0.5) is (10 nCr4_/2^10 = 0.205
Calc: binomp­df(­10,.5,8) = (n,p,r) r is number of successes
Math: (n r) pr(1-­p)n-r

Binomial Random Variables

Binomial with n=15 and p = .4
Compute:
P(X=3)­=bi­nom­pdf­(15­,.4,3)
P(X<=3­)=b­ino­mcd­f(1­5,.4,3)
P(X<3)­=bi­nom­cdf­(15­,.4,2)
P(X>3)­=1-­bin­omc­df(­15,.4,3)
P(X>=3­)=1­-bi­nom­cdf­(15­,.4,2)
P(4<x<­=8)­=bi­nom­cdf­(15­,.4­,8)­-bi­nom­cdf­(15­,.4,4)
P(1.3<­X<1.7)=0

Continuous & Normal Random Variable

For C RV, P(X=3) or any other value is zero.
Probab­ility is area under curve.
Normal RV is bell curve.
Total area under bell is 1.
P(X<a) is the are under the curve up to x=a.
1 Std. Dev. P(-1<Z­<1)= .683
2 Std. Dev. P(-2<Z­<2)­=.954
3 Std. Dev. P(-3<Z­<3)­=.997

Pure Numbers

X is a random normal variable with mean -3 and standard deviation 0.7
P(-4<X­<-3) = normal­cdf­(-4­,-3­,-3,.7)
P(X>-2) = normal­cdf­(-2­,1E­99,­-3,.7)
P(X<=-­3.5­)=n­orm­alc­df(­-1e­((,­-3.5­,-­3,.7)
P(X=-3) =0
P(|X-(­-3)­|>.7­)=­P(X­<-3.7)­+P(­X>-2.3) = normal­cdf­(-1­E99­,-3.7,­-3,.7)­+no­rma­lcd­f(-­2.3­,1E­99,­-3,.7)
A car model gets 24 mpg on the car sticker. The maker knows that this is normally distri­buted with a std dev of 3 mpg. What is the proportion of cars that get less than 20 mpg? P(X,20­)=n­orm­alc­df(­-1E­99,­20,­24,­3)=0.91

Condit­ional Probab­ility

P(A|B) is the probab­ility of A given that B happened.
P(A|B) = P(Aint­ers­ectB) /P(B)
If P(A|B) = P(A) then indepe­ndent.
P(Aint­ers­ectB) = P(A)P(B)
If mutually exclusive P(A|B)=0

Bayes's Theorem

Central Limit Theorem

As n becomes large, the sample mean will be distri­buted according to the normal distri­bution with parameters u and standard deviation - std dev/sqrt n
*As n gets large, the spread in the sample mean distri­bution narrows. This means that the sample mean is more likely to be near the true mean.

>IN­FER­ENTIAL STATIS­TIC­S<

Z-test approx­imated by normal distri­bution. If sample size is large or variance is known.
T-scor­e/test is used when:
- sample size is below 30
- population standard deviation is unknown (estimated from your sample data)
otherwise use z-scor­e/test.
Generally use 95% confidence level.
Z-Stat represents how many std. deviations away from the mean the sample mean is.
Std. dev is std dev/sqrt n
Null hypotheses assumes that whatever you are trying to prove did not happen.
p-value of 0.03 means there is a 3% chance of finding a difference as large as or larger than the one in your study given the null hypothesis is true.
If 0.05 or less you typically do not accept the null hypoth­esis.
Type 1 error: rejecting the null hypothesis when true
Type 2 error: accepting the null hypothesis when false

Two Sample T Test

 

Calculus Notes

 

Deriva­tives and Tangent Lines

The derivative of a function f at x is the slope of the tangent at x. If all of the slopes are assembled you get f'(x) or df/dx.

If we know f'(a) and f(a), the the tangent line at x=a is y=f'(a­)(x­-a)­+f(a)
Slope is f'(a) and line passes through (a,f(a))

Approx­imate slope use: nDeriv

Computing Areas & Integrals

The definite integral of f from a to b is the area underneath the curve from a to b.
Where f is negative, the area contri­buted is a negative area.
Use fnInt

Fundam­ental Theorem of Calculus

Limits

A function f(x) converges to a limit L at x=a if, for any given error tolerance, we can specify a range of x such that for any x in that range, f(x) is near L, near being given the tolerance.

Basic Derivative Rules

Basic Derivative Rules

Basic Derivative Rules

Critical Points

Min: f goes from decreasing to increasing
f' goes from negative to positive.
Max: f goes from increasing to decrea­sing.
f' goes from positive to negative.
Flat: f continues to change in the same way.
f' does not change sign.
f" gives concavity.
Concave up means second derivative is positive which means first derivative is increasing
Concave down: f"<0, f' decrea­sing.

Max/Min Word Problems

10 meters of string. maximum area dimens­ions?
Perimeter: P(l,w) = 2l_2w P=10
Area: A9l,w)=lw
a(l)=l­(5-1)
max is at l=2.5

Newton's Method

1. Pick a, initial guess.
2. Compute tangent line approx­ima­tion: y = f(a)+f­'(a­)(x-a)
3. Solve y=0 and get x = (f'(a)­a-f­(a)­)/f'(a)
4. Use x for the next guess. Repeat.

>In­teg­rals, Series<

Integr­ation by Parts

Integr­ation by Parts

Integr­ation by Parts

Integr­ation by Parts

Arc Length

Taylor Polyno­mials

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Comments

Nice sheet. I just wanted to clarify if your standard deviation formula is really the variance. You would take the square root of variance to get sigma which is standard deviation.

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