stats t1 Flashcards
What is a PMF
the probability a random variable X = x
it is discrete and must sum to 1.
For continuous data a pdf is used
What is a PDF
the probability x falls between the intervals a ands b is the integral F(x) within the limits a and b
What is the cdf for discrete data
The sum of the PMF is taken for all values of K less than or equal to X
what is the cdf. for continuous data
P(X<=x) =. the integral of the pdf, from - infinity to x
what is the expectation of X for discrete data
the sum of xP(X), from x
what is the expectation of X for continuous
the Integral of x*pmf,
from -infinity to infinity
Var(X) = E[{X − E(X)}^2]
= E(X^2) − E(X)^2
What is the Rth moment
E(X^R)
what it the R’th central moment
E[(X- mu)^R] where mu =. E(X)
What is the Co-variance of X and Y
E(XY ) − E(X)E(Y )
What is correlation of X and Y
Cov(X,Y)/root(var(x)var(y))
what are the. linear combination formulas
E(aX+bY+c)=aE(X)+bE(Y)+c
Var(aX + bY + c) = a^2Var(X) + b^2Var(Y ) + 2abCov(X, Y )
X and Y are independent if
fX,Y (x, y) = fX (x)fY (y) or
P(X ≤x,Y ≤y)=P(X ≤x)P(Y ≤y)
for all x and y. If X and Y are independent. then
E(XY) = E(X)E(Y) such that
COV(XY) = Corr(XY) = 0
what is the marginal distribution of X
The marginal distribution of a random variable X is its probability distribution when ignoring any other variables it may be paired with
What is the law of. iterated expectations
E(Y) = E[E(Y |X)]
uniform distribution
P(X=x) = 1/|A| where a is a set
Suitable when all values in A are equally likely.
Bernoulli distribution
X ∼ Ber(θ), where 0 < θ < 1.
P (X = x) = θ^x(1 − θ)^(1−x)
for x = 0, 1
Suitable when there are only two possible values. Note
that θ = P (X = 1).
Binomial distribution
X ∼ Bin(m, θ), where m is a positive inte- ger and 0 < θ < 1.
(MCX)* θ^X(1-θ)^M-X
Suitable when X represents the number of ‘suc- cesses’ in m independent Bernoulli trials and θ is the probability of a success on each trial (or analogous situations)
Geometric distribution
X ∼ Geo(θ), where 0 < θ < 1
P (X = x) = θ(1 − θ)^x
Suitable when X represents the number of failures before the first success in a sequence of indepen- dent Bernoulli trials and θ is the probability of a success on each trial
Poisson distribution
X ∼ Poi(λ), where λ > 0.
P(X=x)= ((λ^x)e^(-λ))/x!
Suitable when X represents the number of events in a fixed time period, where events occur in- dependently and at random with mean rate λ per unit of time
uniform distribution
Uniform. X∼Uni(a,b),where −∞<a<b<∞.
f(x) = 1/b-a
for a<x<b.
Suitable when all values in (a,b) are equally likely, or, more formally, all sub-intervals of a given length are equally likely
exponential
Exponential. X ∼ Exp(λ), where λ > 0.
λe^-λx
Suitable when X represents the time until the first event, where events occur independently and at random with mean rate λ per unit of time
normal
X ∼ N(μ,σ2), where −∞ < μ < ∞ and σ>0.
f(x) = (1/root(2πσ)) * e^-(x−μ)^2(2σ^2)
A symmetric, unimodal distribution on the whole real line. Suitable when X represents the sum or mean of a large number of random values
