probability and probability distribution Flashcards

1
Q

what is the frequentist view ?

A

the proportion of trials in which an outcome occurs, calculated as the no of trials approaches infinity

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2
Q

what is the subjective view ?

A

someones subjective belief about somethings likelihood. however, adjusted in light of evidence.

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3
Q

what is an axiomatic approach ?

A

where you ignore the actual problem and just examine the numbers. there is no mention if fairness

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4
Q

define these probability question terms

a. experiment
b. sample space
c. trial
d. event

A

a. an activity with a range of outcomes
b. all possible outcomes
c. a single performance of the experiment
d. one of the possible outcomes

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5
Q

what is the complement of P(A) ?

A

P(not A)

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6
Q

what are the basic rules of probability ?

A

0 < P(X) < 1

sum of P = 1, for all outcomes

P(not A) = 1 - P(A)

P(A or B) = P(A) + P(B)

P (A and B) = P(A) x P(B)

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7
Q

what is the notation for

a. P(A or B)
b. P(A and B)

A

a. P(AUB)

b. P(ANB)

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8
Q

what does mutually exclusive mean ?

A

two events cannot happen simultaneously

e.g. a marble cannot be square and round

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9
Q

what are independent variables ?

A

when the first event has no effect on the second event’s probability.

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10
Q

what are dependent variables ?

A

when the first event has an effect on the second event’s probability.

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11
Q

when events A and B are not mutually exclusive, how do you calculate P(AUB) ?

A

P(A) + P(B) - P(ANB)

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12
Q

what is

a. marginal probability
b. union probability
c. joint probability
d. conditional probability

A

a. just plain probability of one event by itself P(A)
b. P(AUB)
c. P(ANB)
d. the probability of one event given that another event has already happened P(A|B)

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13
Q

how do you calculate conditional probability ?

A

P(A|B) = P(ANB)/P(B)

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14
Q

how can you tell if two events are independent ?

A

rearrange the conditional probability equation to :
P(ANB) = P(A) . P(A|B)
and the two event are only equal when this is true

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15
Q

to do combination probability we must be able to count the number of outcomes. what are the two types of combination probability, and what do they mean ?

A
  1. combination - order of elements in outcomes doesn’t matter
  2. permutation - order of elements in outcome does matter
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16
Q

how do you calculate the number of : ( combinations with replacement ) ?

A

N^r

n = number of options of elements
r = number chosen
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17
Q

how do you calculate the number of : ( combinations without replacement ) ?

A

nPr = n! / (n-r)!

n = number of options of elements
r = number chosen
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18
Q

how do you calculate the number of : ( permutations with replacement ) ?

A

nCr = (r + n - 1)! / (r)! . (n -1)!

n = number of options of elements
r = number chosen
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19
Q

how do you calculate the number of : ( permutations without replacement ) ?

A

nPr = n! / (r)! . (n-r)!

20
Q

what is bayes theorem ?

A

when there are false positives and false negatives

pregnancy example

Pr(P|+) = Pr(PN+) / Pr(PN+) + Pr(NPN+)

21
Q

what is bayesian updating ?

A

when you have a prior possibility (posterior possibility e.g. probability of being pregnant before the test is even taken)
you use bayes rule to calculate the posterior possibility when there is new info

22
Q

what is the bayes rule ?

A

Pr(A|B) = Pr(ANB) / Pr(ANB) + Pr(NANB)

23
Q

what types of probability distribution are there ?

A
normal 
poisson
binomial
cumulative binomial
uniform
24
Q

what are probability distributions ?

A

each outcome in the sample space has a probability - this displayed is call the probability distribution

  • they can be discrete or continuous.
  • probability and range of outcomes may still be known
  • can take on any value in the potential outcomes
25
what are binomial probability distributions?
they are a family of probability distribution | - two parameters: n , p
26
name the characteristics of a binomial distribution ?
``` . n no of trials . trials are independent . only 2 outcome possibilities . one outcome labeled success . Pr ( success ) = P . Pr ( failure ) = 1 - P . P doesn't change between trials . r = no of successes ```
27
in a binomial distribution what is the Pr( r ) ?
nCr . P^r . (1-p)^(n-r)
28
in a binomial distribution what is the mean ?
E(r) = n x p
29
in a binomial distribution what is the variance ?
v(r) = E[ (r-mean) ^2 ] = sum of (r-mean) ^2 x p = n . p . (1-P)
30
how do you calculate cumulative binomial distribution ?
Pr( r < C ) = Pr( r = 0 ) + ... Pr ( r = c ) or find the statistical table for the distribution and look up the Pr
31
describe a poisson distribution ?
- two possible outcomes - repeated finite no times - p of ' success' is the same every time - it is the probability of very rare event
32
how do you calculate the probability of a success in a poisson ?
P(X) = (mean^X) . ( e^ -mean) / X!
33
compare binomial and poisson distribution ?
- binomial : discrete data, finite support | - poisson : data goes to infinity, no finite support
34
what is uniform distribution ?
continuous data all in one solid block of even probability, you can't calculate the probability of just one event distribution area = 1 and the area goes on the x axis from a to b
35
in a uniform distribution, how do you calculate the probability of each event ?
f(x) = 1/b-a
36
in a uniform distribution, how do you calculate the mean ?
mean = a+b / 2
37
in a uniform distribution, how do you calculate the variance ?
(b - a) ^2 /12
38
in a uniform distribution, how do you calculate the standard deviation ?
(b - a) / ROOT[12]
39
name the types of continuous probability distributions ?
1. normal ( gaussian ) distribution 2. cummulative distribution function 3. linear combinations of random variables
40
describe the normal ( gaussian ) distribution ?
pr sort of (x) = F(X) = 1/ standarddev . rot[2pie] all times by e^ -1/2(x-mean/sd)^2
41
how do you calculate the mean in a normal ( gaussian ) distribution ?
the mean is half of the probability mass
42
where is the majority of the distribution in ?
within 3 sd either side of the mean.
43
where of the x axis of a normal distribution graph is this a. Pr(a > z) b. Pr(a < z) c. Pr(a < z < b)
a. to the right of Z b. to the left of Z c. in between Za and Zb
44
what is f(x) for the cumulative distribution function ?
(1/root{2pie}) x e^(-1/2 )xX^2
45
what is Y for a linear combination of random variables ?
Y = W1.X1 + W2.X2
46
what is the ...... for a linear combination of random variables ? a. expected value ( mean ) b. variance of each c. variance overall
a. same added b. same added c. same times by W^2 added
47
describe joint probabilities?
joint probability Pr ( X=x,Y=y) the sum = 1 the marginal probability would now be the sum of all the options where X=x and y = whatever