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Random Variables
Random Variable
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Random variable:
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is a real-valued function on a sample space
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is denoted by a capital letter
Example:
–
Let X denotes the number of heads obtained in three tosses of
a coin then X is a random variable as below:
X (HHH)=3
X (HHT)=2
X (HTH)=2
X (HTT)=1
.. .
Random Variable
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Random variables can be defined for both continuous and
discrete observations
A discrete random variable has a finite or countably infinite
number of possible values
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Example:
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Discrete: number of head in three tosses of a coin
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Discrete: number of vowels in a word
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Continuous: the area of an apartment
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Continuous: the semantic similarity between two words
Probability Function
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A probability function f over a random
variable X is:
f (x)= P ( X = x) where x is a value of X
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Probability function is useful only for discrete
random variables
Probability Function (p.f.)
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Example: let X denotes the number of heads
obtained in three tosses of a coin then the
probability function of X is:
f (x)=
{
1
8
3
8
3
8
1
8
x =0
x =1
x =2
x =3
Distribution Function
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A distribution function F over a random
variable X is:
F (x)= P ( X ≤ x) where x is a value of X
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Distribution function can be used for both
discrete and continuous random variables
Distribution Function (d.f.)
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Example: let X denotes the number of heads
obtained in three tosses of a coin then the
distribution function of X is:
F (x)=
{
0
1
8
1
2
7
8
1
x<0
0≤ x <1
1≤ x < 2
2≤ x < 3
3≤ x
Properties of Distribution Function
F (−∞)= lim F (x)=0
x →−∞
F (+∞)= lim F ( x)=1
x →+∞
if x < y
then
F ( x)≤ F ( y)
F is continuous from the right at every point x
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NOTE: any function with these properties is the
distribution function of some random variable
Estimating Probability through
Distribution Functions
P (x < X ≤ y)= F ( y)− F (x)
P (x < X < y)= lim F (t )− F ( x)
t→y
−
P (x ≤ X ≤ y)= F ( y)− lim F (t )
t → x−
P (x ≤ X < y)= lim F (t)− lim F (t)
t → y−
t → x−
Probability and Distribution Function
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Example 4.1.3. We toss two dice. Let X be sum of the points
obtained. Then f(x) and F(x) are:
X=
2
3
4
5
6
7
8
9
10
11
12
f(X)
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
X in
(-∞,2)
[2,3)
[3,4)
[4,5)
[5,6)
[6,7)
[7,8)
[8,9)
[9,10)
F(X)
0
1/36
3/36
6/36
10/36
15/36
21/36
26/36
30/36
X in
[10,11)
[11,12)
[12,∞)
F(X)
33/36
35/36
1
Probability and Distribution Function
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Ex. 4.1.2. Let X be the number of heads
obtained in five independent tosses of a fair
coin. Draw a histogram for its probability
function and a graph for its distribution function.
Well Known Discrete Random
Variables
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A Bernoulli random variable X takes two
values {0,1}. The probability of taking 1 is p and
the probability of taking 0 is q=1-p
A Bernoulli experiment is a success/failure
experiment
Example: We toss a die. What is the probability
of obtaining 6?
Well Known Discrete Random
Variables
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A binomial random variable measures the number of successes
in a sequence of n independent Bernoulli experiments.
Example: We toss a die three times. What is the probability of
obtaining two sixes?
If the probability of success in each experiment is p and the
probability of failure is q then the probability of X=x successes out
of n trials is
f (x)= n p x q (n− x )
x
()
if x =0,1,2,. . .
Well Known Discrete Random
Variables
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The values of a discrete uniform random
variable occur with the same probability
Example: We toss a die. What is the probability
of obtaining each of the numbers from one to
six?
1
f ( X = x i )=
n
X ∈{x 1 , x 2 , … , x n }
Well Known Discrete Random
Variables
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The values of a geometric random variable are equal
to the number of independent Bernoulli trials with
parameter p to obtain a success.
We throw a die repeatedly until a six comes up. Let X
be the number of throws. Find the probability function
of X.
P ( X =k )= pq k − 1
where
q=1− p
Exercises
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Ex. 4.1.3
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Ex. 4.1.5
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Ex. 4.1.10
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Ex. 4.1.12