Mathematical Theory of Probability

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A D MATH 4733: Lecture #10 Notes September 19, 2013

Lecture #10 Notes Summary

We covered basic properties of the expectation value, transformations of the uniform distribution,

Bernoulli RVs, and the Binomial Distribution. Its important to note that while the Bernoulli trials and

Binomial distribution are similar, they are NOT the same. The Binomial Distribution can be considered

a compound experiment including the Bernoulli trials.

Topics Covered

Page

Properties of the Expectation Value 1

Properties of Variance 1

Transforming the Uniform Distribution 2

Example 1 3

Bernoulli Random Variables 3

Binomial Distribution 4

Properties of the Expectation Value

1. E[ (x)] = E[(x)] for all R

2. E[(x) + (x)] = E[(x)] + E[(x)]

3. E[] = for all R

Properties of Variance

1. Var(X) = E

X2 E[X]

2

2. Var(X) = 2 Var(X)

3. Var(X + ) = Var(X)

Proof.

Var(X + ) = E|(X + ) E[X + ]|

2

= E|X + E[X] |

2= E

|X E[X]|

2

= Var(X)

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Transforming the Uniform Distribution

Let X be uniformly distributed on [a, b]. This is the same as saying that Y = Xaba

is uniform on [0, 1]. By

definition, X is uniformly distributed on [a, b] if its cdf looks like

To check that Y is uniformly distributed on [0, 1], we need to verify that its cdf is

We just need to use the definitions

Fy(y) = P{Y y} = P{X a

b a y}

= P{X y(b a) + a}

= Fx(a + (b a)y)

=

0, a + (b a)y a(a + (b a)y) a

b a, a a + (b a)y b

1, a + (b a)y b

=

0, y 0

y, 0 y 1

1, y 1

Therefore we have three main results:

1. X = a(b a)Y

2. E[X] = a + (b a)E[Y]

3. Var(X) = (b a)2Var(Y)

Transforming the Uniform Distribution continued on next page. . . Page 2 of 4


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Example 1

Say we wanted to find the expected value of a uniform distribution on [2, 4]. We can first find the expected

value of a uniform distribution on [0, 1] then just transform the results.

E[Y] = R

yp(y) dy = 1

0

y dy =1

2

y21

0

=1

2

E[X] = 2 + (4 2)E[Y] = 2 + 2(1

2) = 3

We can do the same with variance

Var(Y) =

1

0

y2 1 dy

1

2

2=

1

3y31

0

1

4=

1

3

1

4=

1

12

Var(X) = (4 2)2Var(Y) = 4(1

12) =

1

3

Bernoulli Random Variables

Definition. A random variable is of Bernoulli type if and only if X take on exactly two values (commonly

0 and 1 for "failure" and "success," respectively). For shorthand notation, we writeX Ber(p) where p is

the success probability. The is read as "has cdf," "is distributed as," or "has distribution".

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Binomial Distribution

Consider the following experiment. Flip a biased coin n times independently. Let P(coin = 1) = p and

= {strings of length n made out of 0 or 1}. Then we have

P{(01110...1)} = (1 p)nkpk

where k is the number of times 1 appears in the string.

Let X : R with X(string) = number of 1s in that string. Note X takes on the values 0, 1, 2,...,n, and

hence is a discrete random variable. Also note that the number of strings with k 1s is given by m =nk

.

Therefore the probability mass function for X is given by

P(X = k) = P(string1, string2, ..., stringm)

= P(string1) + P(string2) + ... + P(stringm)

= P(string1) + P(string1) + ... + P(string1)

= mP(string1)

= n

k(1 p)nkpk

Definition. X has a binomial distribution with parameters n and p if and only if the outcomes of X are

0, 1, 2, 3,...,n and P(X = k) =nk

pk(1 p)nk where k is the number of successes in n independent trials

with individual probabilities p. The shorthand is X Bin(n, p).

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Mathematical Theory of Probability

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