MA 320-001: Introductory Probability

David Murrugarra

Department of Mathematics,University of Kentucky

http://www.math.uky.edu/~dmu228/ma320/

Spring 2017

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 1 / 15

Section 7.2 Sums of Continuous Random Variables

Definition (Convolutions)Let X and Y be two continuous random variables with densityfunctions f (x) and g(y), respectively. Assume that both f (x) and g(y)are defined for all real numbers. Then the convolution f ∗ g of f and gis the function given by

(f ∗ g)(z) =∫ ∞−∞

f (z − y)g(y) dx =

∫ ∞−∞

g(z − y)f (y) dx

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 2 / 15

Section 7.2 Sums of Continuous Random Variables

Theorem (Theorem 7.1)Let X and Y be two independent random variables with density Let Xand Y be two continuous random variables with density functions f (x)and g(y), respectively. Then the sum Z = X + Y is a random variablewith density function fZ (z), where fZ (z) is the convolution of f (x) andg(y).

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 3 / 15

Section 7.2 Sum of Two Independent Uniform RandomVariables

Example (Example 7.3)Suppose we choose independently two numbers at random from theinterval [0,1] with uniform probability density. What is the density oftheir sum?

Solution:

Let X and Y be random variables describing our choices andZ = X + Y their sum. Then we have

fZ (z) =

z, if 0 ≤ z ≤ 12− z, if 1 ≤ z ≤ 20, otherwise.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 4 / 15

Section 7.2 Sum of Two Independent Uniform RandomVariables

Example (Example 7.3)Suppose we choose independently two numbers at random from theinterval [0,1] with uniform probability density. What is the density oftheir sum?

Solution:

Let X and Y be random variables describing our choices andZ = X + Y their sum. Then we have

fZ (z) =

z, if 0 ≤ z ≤ 12− z, if 1 ≤ z ≤ 20, otherwise.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 4 / 15

Section 7.2 Sum of Two Independent RandomVariables

Example (Homework12: Problem 6)Let X and Y be independent continuous random variables with thefollowing density functions:

fX (x) =

{2x 0 ≤ x ≤ 1,0 otherwise.

fY (y) =

{5y4 0 ≤ y ≤ 1,0 otherwise.

Find the density function, f (z), of X + Y for 0 ≤ z < 1.

Your answer should be a function of z.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 5 / 15

Section 7.2 Sum of Two Independent ExponentialRandom Variables

Example (Example 7.4)Suppose we choose independently two numbers at random from theinterval [0,∞) with exponential density with parameter λ. What is thedensity of their sum?

Solution:

Let X and Y be random variables describing our choices andZ = X + Y their sum. Then we have

fZ (z) =

{λ2ze−λz , if z ≥ 00, otherwise.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 6 / 15

Section 7.2 Sum of Two Independent ExponentialRandom Variables

Example (Example 7.4)Suppose we choose independently two numbers at random from theinterval [0,∞) with exponential density with parameter λ. What is thedensity of their sum?

Solution:

Let X and Y be random variables describing our choices andZ = X + Y their sum. Then we have

fZ (z) =

{λ2ze−λz , if z ≥ 00, otherwise.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 6 / 15

Section 7.2 Sum of Two Independent Normal RandomVariables

Example (Example 7.5)Suppose X and Y are two independent random variables, each withthe standard normal density. What is the density of their sum?

Solution:

We havefX (x) = fY (x) =

1√2π

e−x2/2

Let Z = X + Y . Then we have

fZ (z) =1√4π

e−z2/4

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 7 / 15

Section 7.2 Sum of Two Independent Normal RandomVariables

Example (Example 7.5)Suppose X and Y are two independent random variables, each withthe standard normal density. What is the density of their sum?

Solution:

We havefX (x) = fY (x) =

1√2π

e−x2/2

Let Z = X + Y . Then we have

fZ (z) =1√4π

e−z2/4

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 7 / 15

Section 7.2 Sum of Two Independent Normal RandomVariables

Example (Example 7.5)Suppose X and Y are two independent random variables, each withthe standard normal density. What is the density of their sum?

Solution:

We havefX (x) = fY (x) =

1√2π

e−x2/2

Let Z = X + Y . Then we have

fZ (z) =1√4π

e−z2/4

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 7 / 15

Section 7.2 Independent Trials

We now consider briefly the distribution of the sum of n independentrandom variables, all having the same density function.

If X1,X2, . . . ,Xn are these random variables.

Let Sn = X1 + X2 + · · ·+ Xn be their sum.

Then we havefSn(x) = (fX1 ∗ fX2 ∗ · · · ∗ fXn)(x)

where the right-hand side is an n-fold convolution.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 8 / 15

Section 7.2 Independent Trials

Example (Example 7.9)Suppose the Xi are uniformly distributed on the interval [0,1]. Then

fXi (x) =

{1, if 0 ≤ x ≤ 10, otherwise.

And

fSn(x) =

1

(n − 1)!

∑0≤j≤x

(−1)j(

nj

)(x − j)n−1, if 0 ≤ x ≤ n

0, otherwise.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 9 / 15

Example (Example 7.9)Suppose the Xi are uniformly distributed on the interval [0,1].

Figure: Convolution of n uniform densities.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 10 / 15

Section 7.2 Independent Trials

Example (Example 7.9)Suppose the Xi are normally distributed, with mean 0 and variance 1.Then

fXi (x) =1√2π

e−x2/2

And

fSn(x) =1√2nπ

e−z2/2n

Figure: Convolution of n standard normaldensities.

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 11 / 15

Sum of Independent Normal Random Variables

Example (Homework12: Problem 5)Let Xk be independent and normally distributed with common mean 2and standard deviation 1 (so their common variance is 1.)Compute

P

(−∞ ≤

25∑k=1

Xk ≤ 56.95

)

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 12 / 15

Sum of Two Independent Poisson Random Variables

ExampleSuppose X and Y are two independent random variables, each withpoisson density.

LetfX (x) =

(λr)x

x!e−λr and fY (x) =

(λs)x

x!e−λs

Let Z = X + Y . Then we have

fZ (z) =(λ(r + s))x

x!e−λ(r+s)

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 13 / 15

Sum of Two Independent Poisson Random Variables

Example (Homework12: Problem 2)Let X1 and X2 have Poisson distributions with the same average rateλ = 0.9 on independent time intervals of length 4 and 1 respectively.Find

Prob (X1 + X2 = 3)

to at least 6 decimal places.

Solution:

Let λ = 0.9, r = 4, and s = 1. Then

fZ (3) =(λ(r + s))x

x!e−λ(r+s) =

(0.9)3(5)3

6e−5(0.9)

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 14 / 15

Sum of Two Independent Poisson Random Variables

Example (Homework12: Problem 2)Let X1 and X2 have Poisson distributions with the same average rateλ = 0.9 on independent time intervals of length 4 and 1 respectively.Find

Prob (X1 + X2 = 3)

to at least 6 decimal places.

Solution:

Let λ = 0.9, r = 4, and s = 1. Then

fZ (3) =(λ(r + s))x

x!e−λ(r+s) =

(0.9)3(5)3

6e−5(0.9)

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 14 / 15

Section 8.1 Law of Large Numbers for DiscreteRandom Variables

Theorem (Chebyshev Inequality)Let X be a discrete random variable with expected value µ = E(X ),and let ε > 0 be any positive real number. Then

P(|X − µ| ≥ ε) ≤ V [X ]

ε2

David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 15 / 15