MA 320-001: Introductory Probability - Mathematicsdmu228/ma320/lectures/sec8.1.pdf · Sum of...
Transcript of MA 320-001: Introductory Probability - Mathematicsdmu228/ma320/lectures/sec8.1.pdf · Sum of...
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 8.1 Spring 2017 1 / 11
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
)
Solution:
Let Y =25∑
k=1
Xk . Then Y is N(50,25). We want
P(−∞ ≤ Y ≤ 56.95) = 0.5 + P(0 ≤ Z ≤ 1.39) = 0.9177
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 2 / 11
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
)
Solution:
Let Y =25∑
k=1
Xk . Then Y is N(50,25). We want
P(−∞ ≤ Y ≤ 56.95) = 0.5 + P(0 ≤ Z ≤ 1.39) = 0.9177
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 2 / 11
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 8.1 Spring 2017 3 / 11
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 8.1 Spring 2017 4 / 11
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 8.1 Spring 2017 4 / 11
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 8.1 Spring 2017 5 / 11
Section 8.1 Law of Large Numbers for DiscreteRandom Variables
Example (Example 8.1)
Let X by any random variable with E(X ) = µ and V (X ) = σ2. Then, ifε = kσ, Chebyshev’s Inequality states that
P(|X − µ| ≥ kσ) ≤ σ2
k2σ2 =1k2
Thus, for any random variable, the probability of a deviation from themean of more than k standard deviations is 1
k2 .
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 6 / 11
Section 8.1 Law of Large Numbers for DiscreteRandom Variables
Example (Example 8.1)
Let X by any random variable with E(X ) = µ and V (X ) = σ2. Then, ifε = kσ, Chebyshev’s Inequality states that
P(|X − µ| ≥ kσ) ≤ σ2
k2σ2 =1k2
Thus, for any random variable, the probability of a deviation from themean of more than k standard deviations is 1
k2 .
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 6 / 11
Section 8.1 Law of Large Numbers for DiscreteRandom Variables
Example (Homework13: Problem 1)Let X be a random variable with expected value 4 and variance 9.According to the Chebyshev inequality,
P(|X − 4| ≥ 0.63) ≤
Solution:
P(|X − 4| ≥ 0.63) ≤ 9(0.63)2
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 7 / 11
Section 8.1 Law of Large Numbers for DiscreteRandom Variables
Example (Homework13: Problem 1)Let X be a random variable with expected value 4 and variance 9.According to the Chebyshev inequality,
P(|X − 4| ≥ 0.63) ≤
Solution:
P(|X − 4| ≥ 0.63) ≤ 9(0.63)2
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 7 / 11
Section 8.1 Law of Large Numbers for DiscreteRandom Variables
Theorem (Law of Large Numbers)Let X1,X2, . . . ,Xn be an independent trial process, with finite expectedvalue µ = E(Xi) and finite variance σ2 = V (Xi). Let
Sn = X1 + X2 + · · ·+ Xn.
Then for any positive real number ε > 0,
P(∣∣∣∣Sn
n− µ
∣∣∣∣ ≥ ε)→ 0 as n→∞.
Equivalently,
P(∣∣∣∣Sn
n− µ
∣∣∣∣ < ε
)→ 1 as n→∞.
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 8 / 11
Example (Homework13: Problem 4)Let X1, . . . ,Xn be an independent trials process with common expectedvalue 3 and common variance 17.
What is the exact value of the expected value of the average An?
What is the variance of the average An?
According to the Chebyshev inequality, P(|An − 3| ≥ 0.1) ≤
Using your bound from the Chebyshev inequality, how many trialsare needed so that P(|An − 3| ≥ 0.1) ≤ 0.31?
Solution:
P(|An − 3| ≥ 0.1) ≤ 17n(0.1)2
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 9 / 11
Example (Homework13: Problem 4)Let X1, . . . ,Xn be an independent trials process with common expectedvalue 3 and common variance 17.
What is the exact value of the expected value of the average An?
What is the variance of the average An?
According to the Chebyshev inequality, P(|An − 3| ≥ 0.1) ≤
Using your bound from the Chebyshev inequality, how many trialsare needed so that P(|An − 3| ≥ 0.1) ≤ 0.31?
Solution:
P(|An − 3| ≥ 0.1) ≤ 17n(0.1)2
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 9 / 11
Section 8.2 Law of Large Numbers for ContinuousRandom Variables
Theorem (Chebyshev Inequality)Let X be a continuous 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 8.1 Spring 2017 10 / 11
Section 8.2 Law of Large Numbers for ContinuousRandom Variables
Theorem (Law of Large Numbers)Let X1,X2, . . . ,Xn be an independent trial process with a continuousdensity function f , finite expected value µ = E(Xi), and finite varianceσ2 = V (Xi). Let
Sn = X1 + X2 + · · ·+ Xn.
Then for any positive real number ε > 0,
P(∣∣∣∣Sn
n− µ
∣∣∣∣ ≥ ε)→ 0 as n→∞.
Equivalently,
P(∣∣∣∣Sn
n− µ
∣∣∣∣ < ε
)→ 1 as n→∞.
David Murrugarra (University of Kentucky) MA 320: Section 8.1 Spring 2017 11 / 11