Continuous Random Variables Flashcards
Bell shaped symmetric curve centered at μ and whose shaped is determined by σ^2
Normal Gaussian
x ~ normal ( μ , σ^2 )
for normal distribution z =
z = ( x - μ ) / σ
monitor continuously
RV is T: the waiting time between the next event (between any two events)
exponential (a continuous) RV
T ~ exponential (λ)
T: the waiting time between the next event (between any two events)
To find the probability p(T</>t) integrate from 0 to t the term λe^-(λt)
sample mean (x-bar)
regular average
(x1+x2+…xn) / n
μ for approximately normal RV
refer to binomial μ
μ = n*p
σ^2 for approximately normal RV
refer to binomialσ^2
σ^2 = np(1-p)
For many trials of independent identically distributed random variable with E(x) = μ and variance σ^2, the distribution of the sum approaches normal distribution
Central Limit Theorem:
For many trials of independent identically distributed random variable with E(x) = μ and variance σ^2, the binomial distribution approaches normal distribution
What is a sample sum?
Sample Sum = S-bar = X1 + X2 + X3…Xn
so that E(Sn) = E(X1) + E(X2) + … E(Xn) = n * μ
and Var(Sn) = var(X1) + var(X2) + … var(Xn) = n * σ^2
sample sum normal distribution
Sn ~ N( nμ , n σ^2 )
E(Sn) = nμ
var(Sn) = n σ^2
sample mean normal distribution
X-bar ~ N( μ , σ^2 / n )
E(x-bar) = μ
var(x-bar) = σ^2 / n
probability statement for normal distribution
p(x-bar </> #)
# = x-bar in the z forumla
A statistic that is used to estimate an unknown constant or parameter
point estimator or point estimate (p-hat)
Most often used to evaluate the overall goodness of an estimator
Mean square error = E( ( p-hat - pop.parameter )^2 )