1 probability and statistical inference (9th edition) chapter 5 (part 2/2) distributions of...

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Probability and Statistical Inference (9th Edition)

Chapter 5 (Part 2/2)

Distributions of Functions of Random Variables

November 25, 2015

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Outline

5.5 Random Functions Associated with Normal Distributions

5.6 The Central Limit Theorem

5.7 Approximations for Discrete Distributions

5.8 Chebyshev’s Inequality and Convergence in Probability

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Random Functions Associated with Normal Distributions

Theorem: Assume that X1, X2,…, Xn are independent random variables with distributions N(μ1,σ1

2), N(μ2,σ22),…, N(μn,σn

2), respectively. Then,

n

i

n

iiiii

n

iii ccNXcY

1 1

22

1

, is

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Proof: Recall the mgf of N(,2) is

),( is Y,

2exp

2exp

1

22

1

1

222

1

1

222

1

n

iii

n

iii

n

iii

n

iii

n

i

iiii

n

iixY

ccNTherefore

ct

ct

tctctcMtM

i

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Example 1: If X1 and X2 are independent normal random variables N(µ1,σ1

2) and N(µ2,σ22), respec

tively, then

X1 + X2 is N(µ1+µ2, σ12+σ2

2), and

X1 - X2 is N(µ1-µ2, σ12+σ2

2)


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Example 2: If X1,X2,…,Xn correspond to random samples from a normal distribution N(μ,σ2), then the sample mean

is N(μ,σ2/n)

Proof:

1

1

n

iiXn

X

n,N is X Therefore,

2 since ,

2

22

2

22

2

t

n

tn

tExp

n

tMM

tExptM

t

nn

x


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One important implication of the distribution of is that it has a greater probability of falling in an interval containing μ than does a single sample Xk

The larger the sample size n, the smaller the variance of the sample mean

“Mean” is a constant, but “sample mean” is a random variable

X

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For example, assume that X1, X2,…, Xn are random samples from N(50,16) distribution. Then, is N(50,16/n). The following figure shows the pdf of with different values of n

XX

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Recall:Let Z1, Z2, …, Zn be i.i.d. N(0,1). Then, w =

Z12+ Z2

2+ …+ Zn2 is χ2(n)

Let X1, X2,…, Xn be independent chi-square random variables with k1, k2,…, kn degrees of freedom, i.e., χ2(k1), χ2(k2),…, χ2(kn), respectively. Then, Y=X1+X2+…+Xn is χ2(k1

+k2+…+kn)

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Theorem: Let X1, X2,…, Xn be random samples from the N(μ,σ2) distribution. The sample mean and sample variance are given by

Then,

(a) and are independent

(b) is χ2(n-1)

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Random Functions Associated with Normal Distributions We will accept (a) without proving it Proof of (b):

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14


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It is interesting to observe that

That is, when the actual mean is replaced by the sample mean, one degree of freedom is lost

1 W

and

2

12

2

2

12

2

nisXX

nisX

U

n

i

i

n

i

i

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Central Limit Theorem

It is useful to first review some related theorems Theorem (Sample Mean): Let X1, X2, …, Xn be a sequenc

e of i.i.d. random variables with mean and variance 2. Then, the sample mean

is a random variable with mean and variance 2/n

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Theorem (Strong Law of Large Numbers): Let X1, X2, …, Xn be a sequence of i.i.d. random variables with mean Then, with probability 1,

That is,

(The sample mean converges almost surely, or converges with probability 1, to the expected value)

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Theorem (Strong Law of Large Numbers)(Cont.): This theorem holds for any distribution of the Xi’s This is one of the most well-known results in probability t

heory

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Theorem (Central Limit Theorem): Let X1, X2, …, Xn be a sequence of i.i.d. random variables with mean and variance 2. Then the distribution of

is N(0,1) as

That is,

(convergence in distribution)

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Theorem (Central Limit Theorem)(Cont.): While tends to “degenerate” to zero (Strong La

w of Large Numbers), the factor in

“spreads out” the probability enough to prevent this degeneration

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Theorem (Central Limit Theorem)(Cont.): One observation that helps make sense of this result is t

hat, in the case of normal distribution (i.e., X1, X2, …, Xn are i.i.d. normal), is N(,2/n)

Hence, is (exactly) N(0,1) for each positive value of n

Thus, in the limit, the distribution must also be N(0,1)

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Theorem (Central Limit Theorem)(Cont.): The powerful fact is that this theorem holds for any distri

bution of the Xi’s It explains the remarkable fact that the empirical frequen

cies of so many natural “populations” exhibit a bell-shaped (i.e., normal) curve

The term “central limit theorem” traces back to George Polya who first used the term in 1920 in the title of a paper. Polya referred to the theorem as “central” due to its importance in probability theory

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The Central Limit Theorem and the Strong Law of Large Numbers are the two fundamental theorems of probability

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Central Limit Theorem Example 1 (Normal Approximation to the Uniform Sum Di

stribution (a.k.a. the Irwin-Hall Distribution)): Let Xi, i=1,2,… be i.i.d. U(0,1). Compare the graph of the pdf of Y=X1+X2+…+Xn, with the graph of the N(n(1/2), n(1/12)) pdf

n=2

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n=4

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Central Limit Theorem Example 2 (Normal Approximation to the Uniform Sum Di

stribution (a.k.a. the Irwin-Hall Distribution)): Let Xi, i=1,2,…,10 be i.i.d. U(0,1). Estimate P(X1+X2+…+X10 > 7)

Solution: With

and by the central limit theorem,

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Example 3 (Normal Approximation to the Chi-Square Distribution): Let X1,X2,…,Xn be i.i.d. N(0,1). Then,

is chi-square with n degrees of freedom, with E(Y)=n and Var(Y)=2n

Recall the pdf of Y is

Let

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The pdf of W is given by

Compare the pdf of W and the pdf of N(0,1):

n=20 n=100

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Approximations for Discrete Distributions

The beauty of the central limit theorem is that it holds regardless of the underlying distribution (even discrete)

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Example 4 (Normal Approximation to the Binomial Distribution): X1,X2,… Xn are random samples from a Bernoulli distribution with μ=p and σ2 = p(1-p). Then, Y=X1+X2+…+Xn is binomial b(n,p). The central limit theorem states that

is N(0,1) as n approaches infinity

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Thus, if n is sufficiently large, the distribution of Y is approximately N(np,np(1-p)), and the probabilities for the binomial distribution b(n,p) can be approximated with this normal distribution, i.e.,

for sufficiently large n

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Consider n=10, p=1/2, i.e., Y~b(10,1/2). Then, by CLT, Y can be approximated by the normal distribution with mean 10(1/2)=5 and variance 10(1/2)(1/2)=5/2. Compare the pmf of Y and the pdf of N(5,5/2):

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Example 5 (Normal Approximation to the Poisson Distribution):

Recall the Poisson pmf

where parameter is both the mean and variance of the distribution

Poisson random variable counts the number of discrete occurrences (sometimes called “events” or “arrivals”) that take place during a time-interval of given length

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A random variable having a Poisson distribution with mean 20 can be thought of as the sum Y of the observations of a random sample of size 20 from a Poisson distribution with mean 1. Thus,

has a distribution that is approximately N(0,1), and the distribution of Y is approximately N(20,20)

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Compare the pmf of Y and the pdf of N(20,20):

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Markov’s Inequality

Theorem (Markov’s Inequality): If X is a continuous random variable that takes only nonnegative values, then for any a>0,

The inequality is valid for all distributions (discrete or continuous)

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Proof:

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Intuition behind Markov’s Inequality, using a fair dice (discrete) example:

Then,

…

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Chebyshev’s Inequality

Theorem (Chebyshev’s Inequality): If X is a continuous random variable with mean and variance 2, then for any k>0,

The inequality is valid for all distributions (discrete or continuous) for which the standard deviation exists

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Proof: Since (X-)2 is a nonnegative random variable, we can apply Markov’s inequality (with a=k2) to obtain

Thus,

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Chebyshev’s Inequality (Another Form)

Chebyshev’s Inequality (another form):

Chebyshev’s inequality states that the probability that X differs from its mean by at least k standard deviations is less than or equal to 1/k2

It follows that the probability that X differs from its mean by less than k standard deviations is at least 1-1/k2

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The importance of Markov’s and Chebyshev’s inequalities is that they enable us to derive (sometimes loose but still useful) bounds on probabilities when only the mean, or both the mean and the variance, of the probability distribution are known

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Example 1: If it is known that X has a mean of 25 and a variance of 16, then, a lower bound for P(17<X<33) is given by

and an upper bound for P(|X-25|>=12) is

The results hold for any distribution with mean 25 and variance 16

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