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Random
Variables AND
DISTRIBUTION
FUNCTION
Consider the experiment of tossing a coin twice.
If we are interested in the number of heads that show on
the top face, describe the sample space.
Solution:
S={ HH , HT , TH , TT }
2
1
1
0
Definition (1):
A random variable is a function that associates a real number
with each element in the sample space.
Remark:
We shall use a capital letter, say X, to denote a random
variable and its corresponding small letter, x in this case,
for one of its values
Definition 3.3
If the space of random variable X is countable, then X is
called a discrete random variable.
Definition 3.4
If the space of random variable X is uncountable, then X
is called a continuous random variable.
3.2. Distribution Functions of Discrete Random
Variables
Definition 3.5.
Let 𝑅𝑥 be the space of the random variable X. The
function f : 𝑅𝑥
IR defined by
f(x) = P(X = x)
is called the probability mass function (p m f) of X.
Example 3.5
A pair of dice consisting of a six-sided die and a four-sided
die is rolled and the sum is determined. Let the random
variable X denote this sum.
• Find the sample space,
• the space of the random variable,
• And probability mass function of X.
Answer:
The sample space of this random experiment is given by
{(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
S=
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)}
The space of the random variable X is given by
𝑅𝑋 = {2, 3, 4, 5, 6, 7, 8, 9, 10}.
Therefore, the probability mass function of X is given by
f(2) = P(X = 2) =
𝟏
𝟐𝟒
f(5) = P(X = 5) =
𝟒
𝟐𝟒
f(8) = P(X = 8) =
𝟑
𝟐𝟒
, f(3) = P(X = 3) =
𝟐
𝟐𝟒
, f(6) = P(X = 6) =
, f(9) = P(X = 9) =
, f(4) = P(X = 4) =
𝟒
𝟐𝟒
𝟐
𝟐𝟒
𝟑
𝟐𝟒
, f(7) = P(X = 7) =
𝟒
𝟐𝟒
, f(10) = P(X = 10) =
𝟏
𝟐𝟒
Example 3.6.
A fair
coin is tossed 3 times. Let the random variable X
denote the number of heads in 3 tosses of the coin. Find the
sample space,
the space of the random variable, and the probability density
function of X.
Answer:
The sample space S of this experiment consists of all binary
sequences
of length 3, that is
S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}.
The space of this random variable is given by
𝑅𝑋 = {0, 1, 2, 3}.
Therefore, the probability density function of X is
given by
f(0) = P(X = 0) =
𝟏
8
3
8
f(1) = P(X = 1) =
f(2) = P(X = 2) =
3
8
f(3) = P(X = 3) =
𝟏
8
Theorem 3.1
If X is a discrete random variable with space 𝑅𝑋 and probability
density function f(x), then
(a). f(x) ≥ 0 for all x in, and
(b).
f(x) = 1.
Example 3.7
If the probability of a random variable X with space 𝑅𝑋 = {1, 2, 3, ..., 12} is
given by
f(x) = k (2x − 1),
then, what is the value of the constant k?
Answer:
1=
f(x)
1 =
12
1 k
1=k
(2x − 1)
12
1 (2x
− 1)
1= k 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23
1 = k 144.
1
144
K=
Definition 3.6.
The cumulative distribution function F(x) of a random
variable X is defined as
F(x) = P(X=x)
for all real numbers x.
Theorem 3.2.
If X is a random variable with the space𝑅𝑥 , then
F(X) =
𝑡≤𝑥 𝑓(𝑡)
Example 3.8
If the probability density function of the random variable X
is given by
1
(2x
144
− 1) for x = 1, 2, 3, ..., 12 then
find the cumulative distribution function of X.
Answer:
The space of the random variable X is given by
𝑅𝑋 = {1, 2, 3, ..., 12}.
Then
𝟏
144
F(1) =
𝑡≤1 𝑓(𝑡)= f(1) =
F(2) =
𝑡≤2 𝑓(𝑡)= f(1) + f(2) =
F(3) =
.. ........
.. ........
𝑡≤3 𝑓(𝑡) = f(1) + f(2) + f(3) =
F(12) =
𝑡≤12 𝑓(𝑡)
4
𝟏
3
+
= 144
14𝟒 144
𝟏
3
5
9
+
+ = 144
144 144 144
= f(1) + f(2) + · · · + f(12) = 1.
This part is in the book from pg 45
best of luck