Lecture 24
Agenda
1. Examples from Normal Distribution
2. Beta distribution
3. Moment Generating Function
Example
Mainly two kind of examples are done for normal distribution.
Example 1
Suppose that men’s neck sizes are approximately normally distributed
with a mean of 16.2 inches and variance of 0.81 square inch. Find
the probability that the neck size of a randomly chosen man lies
between 13.5 and 18.9 inhes.
Let X = Men’s neck size in inches. Then X ∼ N(µ = 16.2, σ2 = 0.81).
P (13.5 ≤ X ≤ 18.9) = P (X ≤ 18.9)− P (X ≤ 13.5)
= Φ
(18.9− 16.2
0.9
)− Φ
(13.5− 16.2
0.9
)= Φ (3)− Φ (−3) = 0.997
Example 2
Suppose scores in an exam follow normal distribution with mean 80and standard deviation 5. What’s the minimum score that you should
get to be in the top 10% ?
Let X be the score of a randomly chosen student and suppose you haveto score minimum x to be in the top 10%.
X ∼ N(µ = 80, σ2 = 52)
Then P (X ≤ x) = 0.9.We also know P (X ≤ x) = Φ
(x−µσ
)= Φ
(x−805
). Hence Φ
(x−805
)= 0.9. Now
from computers I can find out that Φ−1(0.9) = 1.2815. Hence x−805
= 1.2815,i.e. x = 86.40776.
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Beta Distribution
Every continuous distribution which we have encountered except the uni-form distribution, takes values over an infinite interval like (0,∞) or R. Thebeta distribution is an alternative model for random variables which can beconstrained in the interval (0, 1). In general if X can be constrained in theinterval (a, b) then Y = X−a
b−a can be constrained in (0, 1) and thus by study-ing properties of Y we can study properties of X.
Definition 1. A random variable X is said to follow the Beta distributionwith parameters (α, β) for some α > 0 and β > 0 if,
Range(X) = (0, 1)
and for x ∈ (0, 1)
fX(x) =Γ(α + β)
Γ(α)Γ(β)xα−1(1− x)β−1
We write this as X ∼ Beta(α, β)
The first thing that we need to check is that∫ 1
0fX(x)dx = 1, i.e. we need
to check the following identity for α, β > 0.∫ 1
0
xα−1(1− x)β−1dx =Γ(α)Γ(β)
Γ(α + β). . . (∗)
but we are omitting this proof for now. Instead let’s do the mean and vari-ance,
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Mean and Variance
E(X) =
∫ 1
0
xfX(x)dx
=
∫ 1
0
x× Γ(α + β)
Γ(α)Γ(β)xα−1(1− x)β−1dx
=Γ(α + β)
Γ(α)Γ(β)
∫ 1
0
xα(1− x)β−1dx
=Γ(α + β)
Γ(α)Γ(β)× Γ(α + 1)Γ(β)
Γ((α + 1) + β)[from (*)]
=Γ(α + β)
Γ((α + 1) + β)× Γ(α + 1)
Γ(α)
=Γ(α + β)
(α + β)Γ(α + β)× αΓ(α)
Γ(α)
=α
α + β
Similarly,
V (X) =αβ
(α + β)2(α + β + 1).
How does the density look like ?
We have plotted the desity for four combinations of α and β. More will bediscussed in lecture. Please note that α = β = 1 means the uniform density.
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Moment generating function
For a random variable X (discrete or continuous), the moment generatingfunction is a special function associated with it. The moment generatingfunction although intuitively looks strange at first is a powerful theoriticaltool. The name “moment generating function” comes from the fact that itcan be used to generate the “moments” of X. We clarify this shortly.
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For any t ∈ R,
E(etX) =
∫ ∞−∞
etxfX(x)dx [If X is continuous]
=∑
x∈Range(X)
etxP (X = x) [If X is discrete]
Now if t 6= 0, there is no guarantee that the above sum or integral will befinite.
Definition 2. Let X be a random variable and A = {t : E(etX) <∞}. Thenwe define the moment generating function of X, as the function MX : A →(0,∞) where for t ∈ A,
MX(t) = E(etX)
Now for any random variable X,
E(X), E(X2), E(X3), . . .
are known as the moments of the random variable. They provide useful in-formation about the random variable. The following result tells us why thisfunction is called the moment generating function.
Lemma 1. If for a random variable X, MX can be defined for all values inany interval (−ε, ε) around 0 then for k ≥ 1,
dk
dtkMX(t)|t=0 = E(Xk)
Thus the moment generating function can be used to generate moments.
Homework::
1. If X ∼ Beta(α, β), then Y = 1−X ∼ Beta(β, α).
2. Prove the formula for variance of beta distribution.
3. 4.123 a, 4.124, 4.125
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