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7/29/2019 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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A D MATH 4733: Lecture #10 Notes September 19, 2013
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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A D MATH 4733: Lecture #10 Notes September 19, 2013
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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A D MATH 4733: Lecture #10 Notes September 19, 2013
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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