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Quantitative Methods Final Exam Cheat Sheet

A cheat sheet in English by rockcollector2
17,950 views 3,566 downloads 10 Aug 2015 History
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Statistics & Probab­ility Notes

Standard Deviation

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)

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

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_/210 = 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

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

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

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

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<

>Integrals, Series<

Integr­ation by Parts

Integration by Parts

Integr­ation by Parts

Integration by Parts

Integr­ation by Parts

Integration by Parts

Integr­ation by Parts

Integration by Parts

Arc Length

Arc Length

Taylor Polyno­mials

Taylor Polynomials

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Comments (1)

abbottmd 17:49 25 Dec 2015

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.

Reply

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