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Random Variables
◦ Learn about population
Aim:
◦ Available information: observed data x1, . . . , xn
Problem:
◦ Data affected by chance variation
◦ New set of data would look different
Suppose we observe/measure some characteristic (variable) of n
individuals. The actual observed values x1, . . . , xn are the outcome
of a random phenomenon.
Random variable: a variable whose value is a numerical outcome of a random phenomenon
Remark: Mathematically, a random variable is a real-valued function on the sample space S:
X
S −−−−→
ω 7−→ x = X(ω)
◦ SX = X(S) is the sample space of the random variable.
◦ The outcome x = X(ω) is called realisation of X.
◦ X induces a probability P (B) =
ability distribution of X
(X ∈ B) on SX , the prob
Example: Roll one die
Outcome ω
Realization X(ω)
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Random Variables
Example: Roll two dice
◦ X1 - number on the first die
◦ X2 - number on the second die
◦ Y = X1 + X2 - total number of points
(a function of random variables is again a random variable)
Table of outcomes:
Outcome (X1 , X2 )
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
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Y
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
Outcome (X1 , X2 )
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
Y
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9
10
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7
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11
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Random Variables
Two important types of random variables:
• Discrete random variable
◦ takes values in a finite or countable set
• Continuous random variable
◦ takes values in a continuum, or uncountable set
◦ probability of any particular outcome x is zero
(X = x) = 0
for all x ∈ SX
Example: Ten tosses of a coin
Suppose we toss a coin ten times. Let
◦ X be the number of heads in ten tosses of a coin
◦ Y be the time it takes to toss ten times
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Discrete Random Variables
Suppose X is a discrete random variables with values x1, x2, . . ..
Example: Roll two dice
Y = X1 + X2 total number of points
y
2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
5
4
3
2
1
(Y = y) 36
36 36 36 36 36 36 36 36 36 36
Frequency function: The function
p(x) = (X = x) = ({ω ∈ S|X(ω) = x})
is called the frequency function or probability mass function.
Note: p defines a probability on SX = {x1 , x2, . . .}:
P
P (B) =
p(x) = (X ∈ B).
x∈B
We call P the (probability) distribution of X.
Properties of a discrete probability distribution
◦ p(x) ≥ 0 for all values of X
P
◦
i p(xi ) = 1
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Discrete Random Variables
Example: Roll one die
Let X denote the number of points on the face turned up. Since
all numbers are equally likely we obtain
½1
if x ∈ {1, . . . , 6}
p(x) = (X = x) = 6
.
0 otherwise
Example: Roll two dice
The probability mass function of the total number of points
Y = X1 + X2
can be written as:
p(y) = (Y = y) =
½
1
36
0
¡
6 − |y − 7|
¢
if y ∈ {2, . . . , 12}
otherwise
Example: Three tosses of a coin
Let X be the number of heads in three tosses of a coin. There are
¡3¢
x outcomes with x heads and 3 − x tails, thus
µ ¶
3 1
p(x) =
.
x 8
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Continuous Random Variables
For a continuous random variable X, the probability that X falls
in the interval (a, b ] is given by
(a < X ≤ B) =
Z
b
f (x)dx,
a
where f is the density function of X.
Note: The density defines a probability on :
¡
¢
¡
¢ Zb
P [a, b] = f (x) dx = X ∈ [a, b]
a
We call P the (probability) distribution of X.
Remark: The definition of P can be extended to (almost) all B ⊆
.
Example: Spinner
Consider a spinner that turns freely on its axis and slowly comes to a stop.
◦ X is the stopping point on the circle marked from 0 to 1.
◦ X can take any value in SX = [0, 1).
◦ The outcomes of X are uniformly distributed over the interval [0, 1).
Then the density function of X is
½
1 if 0 ≤ x < 1
f (x) =
.
0 otherwise
Consequently
¡
¢
X ∈ [a, b] = b − a.
Note that for all possible outcomes x ∈ [0, 1) we have
¡
¢
X ∈ [x, x] = x − x = 0.
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Independence of Random Variables
Recall: Two events A and B are independent if
(A ∩ B) = (A) (B)
Independence of Random Variables
Two discrete random variables X and Y are independent if
(X ∈ A, Y ∈ B) = (X ∈ A) (Y ∈ B)
for all A ⊆ SX and B ⊆ SY .
Remark: It is sufficient to show that
(X = x, Y = y) = pX (x) pY (y) = (X = x) (Y = y)
for all x ∈ SX and y ∈ SY .
More generally, X1 , X2 , . . . are independent if for all n ∈
(X1 ∈ A1 , . . . , Xn ∈ An ) =
(X1 ∈ A1 ) · · · (Xn ∈ An ).
for all Ai ⊆ Xi .
Example: Toss coin three times
Consider
Xi =
½
1
0
if head in ith toss of coin
otherwise
X1 , X2 , and X3 are independent:
(X1 = x1 , . . . , X3 = x3 ) =
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=
8
(X1 = x1 ) (X2 = x2 ) (X3 = x3 )
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Multivariate Distributions: Discrete Case
Discrete Case
Let X and Y be discrete random variables.
Joint frequency function of X and Y
pXY (x, y) = (X = x, Y = y) = ({X = x} ∩ {Y = y})
Marginal frequency function of X
pX (x) =
P
pXY (x, yi)
i
Marginal frequency function of Y
pY (y) =
P
pXY (xi, y)
i
The random variables X and Y are independent if and only if
pXY (x, y) = pX (x) pY (y)
for all possible values x ∈ SX and y ∈ SY .
Conditional probability of X = x given Y = y
(X = x|Y = y) = pX|Y (x|y) =
pXY (x, y)
pY (y)
=
(X = x, Y = y)
(Y = y)
where pX|Y (x|y) is the conditional frequency function.
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Multivariate Distributions
Discrete Case
Example: Three Tosses of a Coin
◦ X - number of heads on the first toss (values in {0, 1})
◦ Y - total number of heads (values in {0, 1, 2, 3})
The joint frequency function pXY (x, y) is given by the following
table
x\y
0
1
2
3
0
1
8
0
1
8
2
8
3
8
0
1
2
8
1
8
3
8
1
8
1
8
1
8
1
2
1
2
1
Marginal frequency function of Y
pY (0) = (Y = 0)
= (Y = 0, X = 0) + (Y = 0, X = 1)
= 81 + 0 =
1
8
pY (1) = (Y = 1)
= (Y = 1, X = 0) + (Y = 1, X = 1)
= 82 + 81 =
...
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Multivariate Distributions
Continuous Case
Let X and Y be continuous random variables.
Joint density function of X and Y : fXY such that
Z Z
A
fXY (x, y) dy dx = (X ∈ A, Y ∈ B)
B
Marginal density function of X:
fX (x) =
Z
fXY (x, y) dy
Marginal density function of Y
fY (y) =
Z
fXY (x, y) dx
The random variables X and Y are independent if and only if
fXY (x, y) = fX (x) fY (y)
for all possible values x ∈ SX and y ∈ SY .
Conditional density function of X given Y = y
fX|Y (x|y) =
fXY (x, y)
fY (y)
Conditional probability of X ∈ A given Y = y
(X ∈ A|Y = y) =
Random Variables, Jan 28, 2003
Z
fX|Y (x|y) dx
A
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