Contents 6 Distributions of Functions of RVs

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ST422 Introduction to Mathematical Statistics II
http://wwwstatncsuedu/people/osborne/courses/st422/
Course Outline:
1. Deriving Distributions of Functions of Random Variables-RVs)
-Ch.6)
Jason A. Osborne
[email protected]
Oﬃce: 5238 SAS Hall
The lecture notes are modiﬁed based on Prof. Tom Gerig’s handouts.
2. Sampling Distributions and the Central Limit Theorem -Ch.7)
3. Estimation and Conﬁdence Intervals -Ch.8)
4. Properties of Point Estimators and Estimation Methods -Ch.9)
5. Hypothesis Testing -Ch.10)
6. Linear Regression -Ch.11, time permitting)
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Contents
Contents
6 Distributions of Functions of RVs
Distributions of Functions of RVs
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Distributions of Functions of RVs
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Let Y be a RV and U = h(Y ) be a RV deﬁned as a function of Y .
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Objective: given the distribution of Y , derive the distribution of U .
6.3
Method of Distribution Functions . . . . . . . . . . . . .
6.4
Method of Transformations . . . . . . . . . . . . . . . . 11
6.5
Method of Moment Generating Functions . . . . . . . . 17
6.7
Order Statistics . . . . . . . . . . . . . . . . . . . . . . 21
Three approaches:
1. Method of distribution functions for continuous RVs):
Integrate over the region in the y space that corresponds to the
event U ≤ u} to obtain the CDF of U : FU (u) = P (U ≤ u).
Diﬀerentiate FU (u) w.r.t. u to obtain the pdf of U : fU (u).
2. Method of transformations for continuous RVs):
Derive the distribution of U by methods akin to the method of
substitution -or change of variable) in calculus
3. Method of moment-generating functions:
Make use of the uniqueness property of mgfs and other properties
to deduce the distribution of functions of RV s.
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Distributions of Functions of RVs
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Method of Distribution Functions
Example 631 (Univariate RV) Let Y = yield (in tons) of pure
sugar per day. Suppose Y has the pdf :
Procedure:
fY (y) = 2y 0 ≤ y ≤ 1.
1. Integrate over the region in the y space that corresponds to the
event U ≤ u} to obtain the CDF of U : FU (u) = P (U ≤ u)
Suppose the daily overhead=100, and the price is 300 per ton of pure
sugar. Daily proﬁt: U = 3Y − 1. Find the distribution of U (i.e. ﬁnd
fU (u)).
2. Diﬀerentiate FU (u) with respect to u to obtain the pdf of U :
fU (u).
Distributions of Functions of RVs
Example 632 (Bivariate RV) Y1 = proportion of gas at the
beginning of a week; Y2 = proportion of gas at the end of a week.
Suppose the joint pdf is
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Distributions of Functions of RVs
Distribution of the Square of a RV
Theorem 1 Let Y be a RV with pdf fY (y) and let U = Y 2 . Then
√ √
1 fU (u) = √ fY ( u) + fY (− u) 0 < u < ∞.
2 u
f (y1 y2 ) = 3y1 0 ≤ y2 ≤ y1 ≤ 1.
Let U = Y1 − Y2 . Determine the pdf of U .
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Distributions of Functions of RVs
Proof:
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Distributions of Functions of RVs
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Example 633 Let Z ∼ N (0 1), and U = Z 2 . Show that
U ∼ χ2 (1).
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Distributions of Functions of RVs
Example 634 Let Y1 Y2 be independent RV s with pdf s:
fY (yi ) = ey 0 ≤ yi < ∞ i = 1 2.
Find the distribution of U = Y1 + Y2 .
Distributions of Functions of RVs
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Method of Transformations
Method of Transformations
Let Y be a RV with pdf fY (y). Let h(Y ) be a -strictly) increasing or
decreasing function of Y , over the range of Y .
A function h(y) is strictly increasing if:
y1 < y2 implies h(y1 ) < h(y2 ).
Objective: ﬁnd the pdf of the RV U = h(Y ).
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Distributions of Functions of RVs
Thus, for h(y) strictly increasing:
dy
fU (u) = fY h1 (u)
du
and for h(y) strictly decreasing:
dy
fU (u) = fY h1 (u) (− ).
du
Then for h(y) strictly increasing or decreasing:
dy fU (u) = fY h1 (u) .
du
Distributions of Functions of RVs
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Example 641 Let Y be a rv with pdf : fY (y) = 3y 2 0 ≤ y ≤ 1.
Distributions of Functions of RVs
Example 643 Let Y ∼ beta(2 2) with pdf :
fY (y) = y(1 − y) 0 ≤ y ≤ 1.
Let U = h(Y ) = θ1 + (θ2 − θ1 )Y , where θ1 < θ2 are parameters.
Derive the pdf of U .
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Example 642 Let Y be a rv with pdf : fY (y) = 3y 2 0 ≤ y ≤ 1.
Let U = h(Y ) = Y 2 . Derive the pdf of U .
Distributions of Functions of RVs
Let U = h(Y ) = 2Y 2 − 1. Derive the pdf of U .
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Distributions of Functions of RVs
Example 644 Let Y ∼ N (µ σ 2 ) with pdf :
fY (y) =
2
2
1
√ eyµ) 2σ ) −∞ < y < ∞.
σ 2π
Let U = h(Y ) = (Y − µ)/σ. Derive the pdf of U .
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Distributions of Functions of RVs
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Distributions of Functions of RVs
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Example 651 Let Y1 Y2 · · · Yn be independent rvs with pdf s:
Method of Moment Generating Functions
fY (yi ) =
Uniqueness Theorem:
Suppose rvs X and Y have mgf s mX (t) and mY (t). If
mX (t) = mY (t) for all t, then X and Y have the same distribution.
1 y β
e
0 ≤ yi < +∞ i = 1 · · · n.
β
Find the distribution of U = Y1 + · · · + Yn . Recall that Yi ∼ Exp(β),
and mY (t) = (1 − βt)1 .
mgf of sums of independent random variables
Let Y1 Y2 · · · Yn be independent rvs with mgf s
mY1 (t) mY2 (t) · · · mYn (t), and let U = Y1 + Y2 + · · · + Yn . Then
mU (t) = mY1 (t) × mY2 (t) × · · · × mYn (t).
Distributions of Functions of RVs
Example 652 Let Y1 Y2 . . . Yn be a random sample from
N (µ σ 2 ). Find the distribution of Y = 1/n(Y1 + · · · + Yn ). Recall
2 2
that for Y ∼ N (µ σ 2 ) mY (t) = exp(µt + t 2σ ).
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Distributions of Functions of RVs
Summary
• Method of distribution function: for continuous RV, general.
• Method of transformation: for continuous RV when h(X) is a
strictly increasing or decreasing function of X.
• Method of MGF: for either discrete or continuous RV. Convenient
for deriving the distribution of a linear combination of independent
RVs.
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Distributions of Functions of RVs
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Distributions of Functions of RVs
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The distribution of the minimum
Order Statistics
The probability density function of the minimum is given by
Let Y1 . . . Yn be a random sample from a population with
CDF F (y) and pdf f (y).
fY1) (y) = n[1 − F (y)]n1 f (y)
Notation: Denote the smallest, or minimum, observation from
Y1 . . . Yn using parenthetical subscripts:
Y1) = min(Y1 Y2 . . . Yn )
Denote the next smallest observation Y2) and so on up until the
largest observation, denoted Yn) so that
Y1) ≤ Y2) ≤ · · · ≤ Yn1) ≤ Yn)
-the order statistics from the sample)
Distributions of Functions of RVs
The distribution of the maximum
The probability density function of the maximum of a random sample
is given by
fYn) (y) = n(F (y))n1 f (y)
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Distributions of Functions of RVs
The distribution of an order statistic in the middle
The pdf of the ith smallest observation is given by
fY) (y) =
n
[F (y)]i1 [1 − F (y)]ni f (y)
(i − 1)(n − i)
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Distributions of Functions of RVs
Example 671 A psychologist studies reaction times to simple
experimental tasks involving memory. Assuming the reaction times for
n = 5 subjects are a random sample from the exponential distribution
with a mean of 7 seconds, ﬁnd P (Y1) < 7) P (Y3) < 7) P (Y5) < 7).
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