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Probability and Statistics (M1710)
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Key Concepts:
Terms in this set (57)
Prior Distribution f(p)
Distribution of P which is fixed before data is collected
Likelihood l(x|p)
Conditional distribution of the experimental outcome given the value of p
posterior distribution f(p|x)
knowledge of p after the experiment
Posterior distribution formula f(p|x)=
l(p|x)(f(p))/f(x)
Union (U)
set of all outcomes belonging to atleast one of events a and b
Intersection
set of all outcomes belonging to both a and b
Mutually exclusive events...
have no outcomes in common
Addition rule for general events
Pr(AuB)=Pr(A) + Pr(B) - Pr(AnB)
permutations
number of ordered arrangements of n
Pr(A|B) =
Pr(AnB)/Pr(B)
If events are independent...
one event having occurred does not affect the likelihood that the other will occur
Two events are statistically independent iff...
Pr(AnB)=Pr(A)*Pr(B)
Total probability theorem
Pr(A)=Sum(Pr(A|Bi)*Pr(Bi))
Bayes' Rule: Pr(B|A)
=Pr(A|B)*Pr(B)/Pr(A)
Probability mass function
the possible values and corresponding probabilities (often shown in a table)
n!/r!(n-r)!
nCr
sum(xi*p(xi))
Expectation of cdf
E(X^2) - [E(X)]^2
Variance of cdf
Cov(X,Y)
E(XY) - E(X)E(Y)
cov(x,y)/sqrt(varx*vary)
cor(x,y)
Bernoulli trial
Independent repeated trials of an experiment with exactly two possible outcomes
Px(X)=(n x)
p^x
(1-p)^n-x
Binomial distribution
np
E[x] of binomial
np(1-p)
Var[x] of binomial
px(x)=(1-P)^x*p
Geometric distribution
(1-p)/p
E[x] of geometric
(1-p)/p^2
Var[x] of geometric
Px(X)=e^-lambda*lambda^x/x!
poisson distribution
lambda
E[x] of poisson
lambda (v)
var[x] of poisson
continuous random variable
can take values anywhere within some interval of the real line
Probability Mass Function (PMF)
Px(X)
Probability Density Function (PDF)
Fx(X)
integralx*fx(X) dx
E[x] of continuous random variable
Fx(X)=lambda
e^-lambda
x
Exponential distribution
1/lambda
E[x] of exponential
1/lambda^2
var[x] of exponential
=integral(0,inf)x^a-1*e^-x dx
Gamma function
Fx(X)=1/(b-a)
Uniform distribution
Cumulative Distribution Function (CDF)'=
probability density function
(a+b)/2
E[x] of uniform
(b-a)^2/12
Var[x] of uniform
x-U(a=0,b=1)
Standard Uniform
Fx(X)=1/b(a,b)
x^a-1
(1-x)^b-1
Beta distribution
a/a+b
E[x] of beta
ab/{(a+b)^2*(a+b+1)}
Var[x] of beta
The normal distribution
let random variable x be normally distributed with parameter u and o^2
mi
nna
1/n-1*sum(xi-x_)^2
variance of a sample population (used in r)
a^2Var(X)+b^2var(y)+2abcov(x,y)
variance for joint distributions
Pr(a)>-0
Pr(sample space)=1
Pr(AuB)=Pr(A)+Pr(B) for mutually exclusive events a and b
3 axioms of probability
Pr(AnB)=pr(A)*Pr(B)
If events are independent
fx(X)=1/sqrt(2pisigma)*exp(-(x-mu)^2/2sigma^2}
Normal distribution
If X-N(mu,sigma^2) then Z =(X-mu)/sigma-N(0,1)
standardization
mu
E[x] of normal
sigma^2
Var[x] of normal
gamma(a)*gamma(b)/gamma(a+b)
B(a,b)
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