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Winter 25 > Stat study > Flashcards

Stat study Flashcards

1
Q

what is pdf for continuous - and what two things about it and two special things

A

using integrals because probability is area
two things, fx is non negative integral from lower to upper bound =1
two special - P(c=x) =0 and no need for < or <=

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2
Q

given pdf for continuous, how to verify is valid

A

check of integral = 1

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3
Q

given pdf for continuous, find probabilities

A

integrate pdf or evaluate cdf

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4
Q

what is cdf and how to find

A

function of probabilities - evaluate using left tailed
find - integrate pdf with respect to y with x bounds - basically integrate but then change to x

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5
Q

how to find pdf from cdf and vs

A

pdf to cdf - integrate
cdf to pdf - derivative

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6
Q

find probabilities given pdf or cdf

A

pdf - integrate with bounds
cdf - evaluate

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7
Q

percentile p of cont, pdf or cdf or standard

A

p = np
integrate pdf from low to np = p and solve for np
evaluate Fnp = p, and solve for np
phinp = p - look in table for p and corresponding np

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8
Q

expected value of cont

A

formula
integrate pdf of xfx - x can be a function
ecx = cex and by adding too affected

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9
Q

variance of cont

A

formula
ex^2-(ex)^2 - x can be a formula too
find expected values
vcx = c^2vx, only affected by scalar

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10
Q

standard deviation of cont

A

square root of variance
only affected by a scalar

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11
Q

given pdf, find a k such that pdf is valid

A

set integral of bounds of fx = 1 and solve for k

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12
Q

given or statements with probabilities

A

add those probabilities

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13
Q

uniform distribution - use and how

A

formulas
intervals

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14
Q

given uniform dis, find pdf

A

use formula

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15
Q

given uniform dis, find probs

A

integrate pdf with bounds

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16
Q

given uniform dis, find ex and ex

A

formulas

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17
Q

given uniform dis, but asking something about binomial (fail or success)

A

identify new var and solve for its probs

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18
Q

normal dis - use and how and few points (2) what u and sqv do to it

A

tell you normal
standardize to use table
symmetric about mean
spread determined by standard deviation
u is location and sqv is scale

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19
Q

percentiles and normal distributions

A

find phinp = p, find np by looking for p in the table, unstandardized np

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20
Q

standard normal dis - use and how and few things about

A

tells us
mean is 0 and sqv is 1
use phi and table
all left tailed

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21
Q

two things to know about phi and complicated probs with it

A

phi(-c) = 1 - phi(c) by symmetry
p(|z|>=c) = p(z<-c) + p(z>c)

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22
Q

going over or under the standard normal table

A

less - 0
more - 1

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23
Q

given standard dis, find probs

A

get into left tailed, use phi and table

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24
Q

given standard dis, find p percentile

A

phi np = p
look in table for p that is np

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25
how to standardized normal
use formula
26
given normal dis, find probs
standardized and then use table
27
given normal dis, find p percentiles
phi np = p look in table for p, that is np value then unstandardized
28
empirical rule
is dis is approx normal 68% of values are 1 standard deviation from mean 95% of values are 2 standard deviations from mean 99.7% of values are 2 standard deviations from mean
29
normal approx of binomial approx use
need to calculate too many binomials (there is a probs and a failure or success) good when np>=5 and p<=.5 very goof when np>=10 and n(1-p) >=10 use continuity equation - draw it out!!
30
given a huge binomial
standardized using equation, careful of bounds!!
31
exponential dis and how to use
formulas told skewed dis to right scale parameter lamba which tells you how fast the values decrease
32
memoryless property
only with exponential initial times does not affect probability
33
given exponential dis, probs
careful with lamba use pdf or cdf formulas
34
given an exponential dis and there is a failure or success thing
make a binomial variable and solve for its probs using the exponential stuff
35
given something has already worked for a time
to is initial time and t is additional use formula and then use exponential formulas to solve for probs
36
gamma dis - use and parameters and gamma function and when to use pretty cdf wanna be
waiting times alpha - shape, <1 decreasing, >1 increasing and then decreasing beta - scale - spread of pdf gamma function - Ta = integral from 0 to inf of x^(a-1)e^-x or alpha is pos int then Ta = (a-1)! pretty - a is pos int - summation from 0 to n-1 of (x/B)^i/i! times e^-x/B
37
given gamma dis, expected value, variance?
formulas
38
probabilities of gamma dis
if alpha is a pos int - use pretty function alpha is not a pos int - integrate dis formula (know gamma function)
39
integrating by parts
integral of f g prime = fg|bounds - integral of f prime g
40
know graphs of lnx, -lns, e^x, -e^x, e^-x
ln x- wanna be sqx -lnx - lnx flipped over x e^x - wanna be half pos x^2 -e^x- e^x flipped over x e^-x - e^x flipped over y
41
two discrete joint pmf facts
pxy =PX=x,PY=y table if all probs are pos and adding up to 1
42
two discrete marginal pmfs
fixing one var and varying the other (of x = summation over ally with that x) in table - adding rows and columns
43
given joint pmf and the x and y values, find c it is try
do the summation and solve for c
44
given joint pmf table and a p function, find probs and marginals
use function by plugging in or table directly varying one and fixing one, summation
45
two discrete independent vars
is for all of x, pxy = marx times mary
46
given pmf table and events
like a puzzle - union and intersection rules!
47
two discrete conditional dis and independence
formulas if independent - px given y = mar x
48
given pxy, px, py, find px|y, px|y=#, px=#|y=#
px|y - formulas...get a function px|y=# - plug y into px|y and get a function px=#|y=# - plug x and y into px|y and get a number
49
two cont joint pdfm when to be true
formula fxy >0 and the integral = 1
50
two cont marginal pdf
formula take integral of fxy wrt to y and bounds of x
51
two cont independent vars
fx times fy = fxy...formula bounds tells you too
52
two cont conditional dis...independence?
joint /given...formulas if independent, fx|y = fx
53
given fxy of a cont rx, find a k that makes it valid
take integral and = 1, solve for k with correct matching of dy dx and bounds
54
probabilities of two cont rvs
integral with right vars more complicated (x from 0 to 2 and y from 2 to 4 - find p of x+y < 3 so use x from 0 to 1 and y from 2 to 3-x)
55
marginals of two cont rvs
use formulas...get a functions
56
conditionals of two cont rvs vary var | not vary
find conditional function based on variables order plug in not varying into function and then integrate with varying variable bounds and wrt
57
expected value of a conditional statement
take integral with bounds of want ..of want var times conditional function with given plugged in wrt to want
58
expected values of two dis or two cont
formulas E[x] = summation of x px (x can be a function and px depends when using one or two vars E[x] = integral of x times fx ( x can be a function and px depends when using one or two vars)
59
1. given joint pmf - find ex,ey and exy, independence 2. given joint pdf and marginals dis - find ex, ey,exy
formulas but imagine it!
60
covariance
formulas independent -> cov = 0 >0 - pos relationship, increase increase <0 - neg relationship, increase decrease
61
correlation
formulas between -1 and 1 independent -> corr = 0 and uncorrelated >0 - pos linear relationship <0 - neg linear relationship += 1 - perfect linear relationship |corr| >= .8 - strong linear relationship .5<=|corr|< .8 - moderate linear relationship |corr|<.5 - weak linear relationship
62
when you have a variable that depends on another...
integrate with the dependent one first
63
given two discrete or two cont..find cov and corr
imagine it
64
distribution of sample mean
formulas x bar is centered at u and variance of x bar decrease as n increases
65
normal case
tells you it is a normal distribution - use for any n - standardized and solve
66
central limit thm
random sample with n>= 30 - bigger n, better approx
67
standardizing random samples
formulas
68
probability of a sample mean
told normal or n>=30, then standardized using "new" u and deviation/denominator and then use phi and table
69
when asking for totals of random samples
use other formulas for random samples
70
percentiles wanna be for random samples
find the right z and the use the right formulas to unstandardized

Winter 25 (2 decks)

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