probability and probability distribution Flashcards
what is the frequentist view ?
the proportion of trials in which an outcome occurs, calculated as the no of trials approaches infinity
what is the subjective view ?
someones subjective belief about somethings likelihood. however, adjusted in light of evidence.
what is an axiomatic approach ?
where you ignore the actual problem and just examine the numbers. there is no mention if fairness
define these probability question terms
a. experiment
b. sample space
c. trial
d. event
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
what is the complement of P(A) ?
P(not A)
what are the basic rules of probability ?
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)
what is the notation for
a. P(A or B)
b. P(A and B)
a. P(AUB)
b. P(ANB)
what does mutually exclusive mean ?
two events cannot happen simultaneously
e.g. a marble cannot be square and round
what are independent variables ?
when the first event has no effect on the second event’s probability.
what are dependent variables ?
when the first event has an effect on the second event’s probability.
when events A and B are not mutually exclusive, how do you calculate P(AUB) ?
P(A) + P(B) - P(ANB)
what is
a. marginal probability
b. union probability
c. joint probability
d. conditional probability
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)
how do you calculate conditional probability ?
P(A|B) = P(ANB)/P(B)
how can you tell if two events are independent ?
rearrange the conditional probability equation to :
P(ANB) = P(A) . P(A|B)
and the two event are only equal when this is true
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 ?
- combination - order of elements in outcomes doesn’t matter
- permutation - order of elements in outcome does matter
how do you calculate the number of : ( combinations with replacement ) ?
N^r
n = number of options of elements r = number chosen
how do you calculate the number of : ( combinations without replacement ) ?
nPr = n! / (n-r)!
n = number of options of elements r = number chosen
how do you calculate the number of : ( permutations with replacement ) ?
nCr = (r + n - 1)! / (r)! . (n -1)!
n = number of options of elements r = number chosen
how do you calculate the number of : ( permutations without replacement ) ?
nPr = n! / (r)! . (n-r)!
what is bayes theorem ?
when there are false positives and false negatives
pregnancy example
Pr(P|+) = Pr(PN+) / Pr(PN+) + Pr(NPN+)
what is bayesian updating ?
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
what is the bayes rule ?
Pr(A|B) = Pr(ANB) / Pr(ANB) + Pr(NANB)
what types of probability distribution are there ?
normal poisson binomial cumulative binomial uniform
what are probability distributions ?
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
what are binomial probability distributions?
they are a family of probability distribution
- two parameters: n , p
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
in a binomial distribution what is the Pr( r ) ?
nCr . P^r . (1-p)^(n-r)
in a binomial distribution what is the mean ?
E(r) = n x p
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)
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
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
how do you calculate the probability of a success in a poisson ?
P(X) = (mean^X) . ( e^ -mean) / X!
compare binomial and poisson distribution ?
- binomial : discrete data, finite support
- poisson : data goes to infinity, no finite support
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
in a uniform distribution, how do you calculate the probability of each event ?
f(x) = 1/b-a
in a uniform distribution, how do you calculate the mean ?
mean = a+b / 2
in a uniform distribution, how do you calculate the variance ?
(b - a) ^2 /12
in a uniform distribution, how do you calculate the standard deviation ?
(b - a) / ROOT[12]
name the types of continuous probability distributions ?
- normal ( gaussian ) distribution
- cummulative distribution function
- linear combinations of random variables
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
how do you calculate the mean in a normal ( gaussian ) distribution ?
the mean is half of the probability mass
where is the majority of the distribution in ?
within 3 sd either side of the mean.
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
what is f(x) for the cumulative distribution function ?
(1/root{2pie}) x e^(-1/2 )xX^2
what is Y for a linear combination of random variables ?
Y = W1.X1 + W2.X2
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
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
