Download Lecture 4

Random Variables
Example
In an opinion poll, 50 people are sampled to ask if they agree
with certain issue. If you record a ’1’ for agree and ’0’ for
disagree. The sample space contain 250 sample points
S = {101 · · · 000
010 · · · 000
···}
We might only interested in the number of people who agree
X = Number of 1s recorded out of 50.
Random variable
I
Definition: A random variable is a function from a sample
space S into real numbers.
I
A discrete random variable is a real-valued function on
discrete sample space.
I
Discrete random variables can take finite or countable
infinite number of values.
Example 1: Tossing 3 fair coins
Suppose that an experiment consists of tossing 3 fair coins. If
we let Y denote the number of heads that appear, then Y is a
random variable taking on one of the values 0,1,2 and 3 with
respective probabilities
P(Y = 0) = P({T , T , T }) =
1
8
3
8
3
P(Y = 2) = P({T , H, H}, {H, T , H}, {H, H, T }) =
8
1
P(Y = 3) = P({H, H, H}) =
8
P(Y = 1) = P({T , T , H}, {T , H, T }, {H, T , T }) =
Example 2: Tossing two dice
Define X (s) = Sum of the numbers of two dice. Then X (s) is a
random variable on sample space S.
S = {s : 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26,
31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46,
51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66}.
s
11
12
13
14
···
65
66
X (s)
2
3
4
5
···
11
12
Example continued
1
36 .
I
PX (X (s) = 2) = P({s : X (s) = 2}) = P({11}) =
I
PX (X (s) = 4) = P({s : X (s) = 4}) = P({13, 31, 22}) =
3
36 .
x
2
3
4
5
6
7
···
11
12
PX (X (s) = x)
1
36
2
36
3
36
4
36
5
36
6
36
···
2
36
1
36
We call PX induced probability function on χ. We can verify that
PX is a probability measure on χ.
Probability mass function (pmf)
I
The probability mass function for X is the function f
defined on the real line by
f (x) = PX (X = x) = P(s : X (s) = x).
P
I
Two properties: (i) f (x) ≥ 0 for all x and (ii)
I
We will simplify PX to P when there is no confusion. We
x
f (x) = 1.
will always use capital letters to denote random variables
and lower case letters to represent the realized value of the
random variable.
Example: pmf for binomial random variables
Suppose you toss a coin 5 times. Let
X = {the number of heads obtained}.
For example:
{s : X (s) = 2} = {HHTTT, HTHTT, HTTHT, HTTTH, THHTT,
THTHT, THTTH, TTHHT, TTHTH, TTTHH}.
Assume (a) the trials are independent and (b) the probabilities
of getting head on each trial are the same. Then for each of the
outcomes in {X = 2}, the probability is (1/2)2 (1/2)3 .
5
2
3
P(X = 2) = 10 × (1/2) (1/2) =
(1/2)2 (1/2)3 .
2
Binomial distribution
In general, assume n experiments are performed, for each trial,
the probability of getting head (‘success’) is p.
n k
f (k ) = P(X = k ) =
p (1 − p)n−k , for k = 0, 1, 2, · · · , n.
k
Then X is said to have binomial distribution with parameters n
and p, having pmf f (k ) for k = 0, 1, · · · , n. If n = 1, X is said to
have Bernoulli distribution with parameter p.
n
X
k=0
n X
n k
f (k ) =
p (1 − p)n−k = (p + (1 − p))n = 1.
k
k=0
Example: pmf for Geometric distribution random
variables
Suppose we toss a coin until a head appear. Let
X =number of tosses required to get a head
p =probability of a head on each toss.
Since the toss are independent and X = k means that we get
k − 1 tails before we get a head on the k−th trial,
f (k ) = P(X = k) = (1 − p)k−1 p,
We can see that
∞
∞
X
X
f (k ) =
(1 − p)k−1 p =
k=1
k=1
k = 1, 2, 3, · · · .
p
= 1.
1 − (1 − p)
Cumulative Distribution Function (CDF)
Definition: The cumulative distribution function or CDF of a
random variable X , denoted by FX (x) is defined by
FX (x) = PX (X ≤ x) for all x.
A random variable is discrete if FX (x) is a step function of x.
Example: Tossing two dice
Let X = sum of the numbers of two dice.


0, −∞ < x < 2;





 1 , 2 ≤ x < 3;


 36
3
FX (x) =
36 , 3 ≤ x < 4;



..


.





1, 12 ≤ x < ∞.
CDF plot
The CDF plot is as follows:
CDF function
The function F (x) is a CDF if and only if the following three
conditions hold
(a) limx→−∞ F (x) = 0 and limx→∞ F (x) = 1.
(b) F (x) is a non-decreasing function of x.
(c) F (x) is right-continuous. i.e, for every number x0 ,
limx↓x0 F (x) = F (x0 ).
Example: CDF for geometric random variable
FX (x) = P(X ≤ x) =
[x]
X
P(X = i)
i=1
 P
[x]
i
[x]

i=1 (1 − p) p = 1 − (1 − p) , if x ≥ 1;
=
 0,
if x < 1.
where [x] is the largest integer no larger than x.