Download MATH20802: Statistical Methods Semester 2 Formulas to remember

MATH20802: Statistical Methods
Semester 2
Formulas to remember for the final exam
The moment generating function of a random variable X is MX (t) = E [exp(tX)].
(0 n)
The fact that E (X n ) = MX (0).
The moment generating function of a Γ(a, λ) random variable is MX (t) =
λ
λ−t
a
.
θb is an unbiased estimator of θ if E θb = θ.
θb is an asymptotically unbiased estimator of θ if lim E θb = θ.
n→∞
The bias of θb is E θb − θ.
The mean squared error of θb is E
θb − θ
θb is a consistent estimator of θ if lim E
2 .
n→∞
θb − θ
2 = 0.
If X ∼ U ni[0, θ] then E (X) = θ/2, Var(X) = θ2 /12 and F (x) = x/θ.
Z ∞
The gamma function is defined by Γ(a) =
ta−1 exp(−t)dt.
0
The fact that Γ(n) = (n − 1)!.
"
The probability density function of X ∼ N µ, σ
2
#
1
(x − µ)2
is fX (x) = √
exp −
.
2σ 2
2πσ
"
The probability density function of X ∼ LN µ, σ
2
#
(log x − µ)2
1
exp −
.
is fX (x) = √
2σ 2
2πσx
If X ∼ LN µ, σ 2 then log X ∼ N µ, σ 2 .
If X ∼ N µ, σ 2 then E(X) = µ and Var(X) = σ 2 .
If X1 ∼ N µ1 , σ12 and X2 ∼ N µ2 , σ22 are independent then aX1 +bX2 ∼ N aµ1 + bµ2 , a2 σ12 + b2 σ22 .
If X1 , X2 , . . . , Xn is a random sample from N µ, σ 2 then X =
n
1X
Xi ∼ N µ, σ 2 /n .
n i=1
The cumulative distribution function of X ∼ N (0, 1) is Φ(x).
The Type I error of H0 : µ = µ0 versus H0 : µ 6= µ0 occurs if H0 is rejected when in fact
µ = µ0 .
The Type II error of H0 : µ = µ0 versus H0 : µ 6= µ0 occurs if H0 is accepted when in
fact µ 6= µ0 .
The significance level of H0 : µ = µ0 versus H0 : µ 6= µ0 is the probability of type I error.
The power function of H0 : µ = µ0 versus H0 : µ 6= µ0 is Π(µ) = Pr (RejectH0 | µ).
If X1 , X2 , . . . , Xn is a random sample from N µ, σ 2 , where σ 2 is assumed known, then
the rejection region for H0 : µ = µ0 versus H1 : µ 6= µ0 is
√ n
X − µ0 > zα/2 .
σ
1
If X1 , X2 , . . . , Xn is a random sample from N µ, σ 2 , where σ 2 is assumed known, then
the rejection region for H0 : µ = µ0 versus H1 : µ < µ0 is
√ n
X − µ0 < −zα .
σ
Based on a random sample X1 , X2 , . . . , Xn from a distribution with the probability
density function f (x; θ), the Neyman-Pearson test rejects H0 : θ = θ1 versus H1 : θ = θ2
if
n
Y
L (θ1 )
= i=1
n
Y
L (θ2 )
f (Xi ; θ1 )
<k
f (Xi ; θ2 )
i=1
for some k.
2