Download STA 4032 Midterm 2 Formula Sheet Chapter 4. Continuous

STA 4032 Midterm 2 Formula Sheet
Chapter 4. Continuous Distributions
Let X be a continuous random variable with probability density function f (x). The cumulative
distribution function of X is defined by
Z x
F (x) = P (X ≤ x) =
f (y)dy.
−∞
We Rhave the following formulae:
∞
(1) −∞ f (x)dx = 1.
Rb
(2) P (a < X ≤ b) = a f (x)dx = F (b) − F (a).
(3)
Z
∞
µ = E(X) =
xf (x)dx.
−∞
(4)
2
2
Z
∞
σ = V (X) = E[(X − E(X)) ] =
(x − µ)2 f (x)dx
−∞
(5)
σ 2 = E(X 2 ) − µ2 .
(6) For any function h,
Z
∞
E[h(X)] =
h(x)f (x)dx
−∞
Some Special Continuous Distributions:
(1) Uniform distribution: p.d.f. f (x) = 1/(b − a), a ≤ x ≤ b; Otherwise, f (x) = 0.
(2) Let X have a normal distribution N (µ, σ 2 ). Then, E[X] = µ and V ar(X) = σ 2 . Let Z =
(X − µ)/σ. Then, Z has a standard normal distribution N (0, 1) with µ = 0 and σ = 1. The
distribution function of X can be expressed by
X −µ
a−µ
FX (a) = P (X ≤ a) = P
≤
σ
σ
a−µ
a−µ
= P Z≤
=Φ
,
σ
σ
where Φ is the distribution function of Z. We also have
b−µ
a−µ
P (a ≤ X ≤ b) = F (b) − F (a) = Φ
−Φ
.
σ
σ
Normal Approximation to the Binomial Distributions
p
If X is a binomial random variable with parameters n and p, then Z = (X − np)/ np(1 − p) is
approximately a standard normal random variable. Continuity correction:
!
x + 0.5 − np
P (X ≤ x) = P (X ≤ x + 0.5) ≈ P Z ≤ p
np(1 − p)
!
x − 0.5 − np
P (X ≥ x) = P (X ≥ x − 0.5) ≈ P Z ≥ p
np(1 − p)
1
(3) Exponential distribution with p.d.f. f (x) = λe−λx , 0 ≤ x < ∞ (λ > 0). The cumulative
probability distribution function is given by
1 − e−λx , x ≥ 0
F (x) =
0,
otherwise.
The expected value and variance of X are
E(X) =
1
,
λ
V (X) =
1
.
λ2
It satisfies the memoryless property, for positive s and t,
P (X > s + t|X > t) = P (X > s).
(4) Weibull distribution: its p.d.f. is defined by
β x β−1
x β
f (x) =
exp −
, 0 ≤ x < ∞,
δ δ
δ
with scale parameter δ > 0 and shape parameter β > 0. Its cumulative distribution function is
β
F (x) = P (X ≤ x) = 1 − e−( δ ) .
x
Its mean µ = E(X) = δΓ(1 + 1/β) and variance
2
σ 2 = V (X) = δ 2 Γ(1 + 2/β) − δ 2 [Γ(1 + 1/β)] .
Chapter 5. Joint Probability Distributions (Sections 5-1 and 5-4)
5-1. Two or More Random Variables
The random variables X and Y are said to be jointly continuous if there is a nonnegative function
fXY (x, y), called the joint probability density function, such that
(1) fXY (x, y) ≥ 0 for all x, y
(2)
Z
Z
∞
∞
−∞
−∞
fXY (x, y)dxdy = 1
(3) For any region R in the xy-plane,
Z Z
P ((X, Y ) ∈ R) =
fXY (x, y)dxdy
R
The marginal density functions of X and Y are given by
Z ∞
Z
fX (x) =
fXY (x, y)dy, fY (y) =
−∞
∞
fXY (x, y)dx
−∞
Conditional Distributions
If X and Y are jointly continuous with joint density function fXY (x, y), then the conditional probability density function of Y given that X = x is given by
fY |x (y) =
fXY (x, y)
fX (x)
2
and the conditional probability density function of X given that Y = y is given by
fXY (x, y)
.
fY (y)
fX|y (x) =
Some related formulas:
Rb
(1) P (a < X < b|Y = y) = a fX|y (x)dx.
Rb
(2) P (a < Y < b|X = x) = a fY |x (y)dy.
R∞
(3) µY |x = E(Y |x) = −∞ yfY |x (y)dy.
R∞
R∞
(4) V (Y |x) = −∞ [y − E(Y |x)]2 fY |x (y)dx = −∞ y 2 fY |x (y)dy − µ2Y |x ..
Independent Random Variables
Two continuous random variables X and Y are independent if fXY (x, y) = fX (x)fY (y).
5-4. Linear Functions of Random Variables
Given random variables X1 , X2 , . . . , Xp and constants c1 , c2 , . . . , cp ,
Y = c1 X1 + c2 X2 + · · · + cp Xp
is a linear combination of X1 , X2 , . . . , Xp .
Some related formulas:
(1) E(Y ) = c1 E(X1 ) + c2 E(X2 ) + · · · + cp E(Xp ).
(2) If X1 , X2 , . . . , Xp are independent, then
V (Y ) = c21 V (X1 ) + c22 V (X2 ) + · · · + c2p V (Xp ).
(3) For sample mean X̄ = (X1 + X2 + · · · + Xp )/p with E(Xi ) = µ for i = 1, 2, . . . , p, we have
E(X̄) = µ. If X1 , X2 , . . . , Xp are also independent with V (Xi ) = σ 2 for i = 1, 2, . . . , p, we have
V (X̄) =
σ2
.
p
(4) If X1 , X2 , . . . , Xp are independent, normal random variables with E(Xi ) = µi and V (Xi ) = σi2
for i = 1, 2, . . . , p, then Y = c1 X1 + c2 X2 + · · · + cp Xp is a normal random variable with
E(Y ) = c1 µ1 + c2 µ2 + · · · + cp µp
and
V (Y ) = c21 σ12 + c22 σ22 + · · · + c2p σp2 .
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