Random Variables
It is easier to work with numbers that with elementary outcomes. From now on,
we’re going to use real-valued functions called random variables (r.v.’s) instead
of working with Ω and ω directly. Random variables allow us to summarize
elementary outcomes and events with numbers.
Definition: A random variable (r.v.) is a function
X : Ω → R.
Example:
Throw a dart at a board with 8 grey squares and 1 red square three times. The
tosses are iid with P (red) = 19 on any toss. Count the number of times you hit
the red square.
• Let X = # of reds.
• X is a random variable.
– X = x if there are x reds and 3 − x greys.
– The set of possible values for X is {0, 1, 2, 3}.
• What is the probability that you hit the red square exactly twice?
– The event “exactly 2 reds” is formally written as
{ω : X(ω) = 2}
– Simpler notation for the event “exactly 2 reds” is X = 2.
P (2 reds) = P (X = 2)
= P (RRG or RGR or GRR)
= P (RRG) + P (RGR) + P (GRR)
1 1 8 1 8 1 8 1 1
· · + · · + · ·
=
9 9 9 9 9 9 9 9 9
• Standard Notation:
– Capital letters for r.v.’s – i.e., X, Y, Z – and lowercase letters for the
values of the r.v.’s
– To simplify, write
X = x for{ω : X(ω) = x}
Example (Practice with notation):
Send 8 bits of information through a communication channel. Each bit
is received correctly with probability p and incorrectly with probability q.
The bits are independent. We are interested in the number of bits that are
received incorrectly.
– Sending the 8 bits is a sequence of iid Bernoulli trials.
– Let X =# of bits received incorrectly.
– What are the possible values for X?
{0, 1, 2, 3, 4, 5, 6, 7, 8}
– Write the following events and expressions for their probabilities using
the r.v. X.
(a.) No wrong bits:
X = 0,
P (X = 0)
(b.) At least one wrong bit:
X > 0,
P (X > 0) or X ≥ 1,
P (X ≥ 1)
(c.) Exactly 2 wrong bits:
X = 2,
P (X = 2)
(d.) Between 2 and 7 wrong bits (inclusive):
2 ≤ X ≤ 7,
P (2 ≤ X ≤ 7)
Definition: The image of a r.v. is the range of the r.v.
Im(X) = R(X) = X(Ω) = {x : x = X(ω) for some ω ∈ Ω}
The image of a r.v. X is sometimes called the sample space for X.
Definition: A r.v. is discrete if Im(X) is finite or countably infinite.
Examples:
Consider following experiments. What is the image? Is the r.v. discrete?
(a.) X =total # dots showing on two rolls of a six-sided die.
Im(X) = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
X is discrete.
(b.) Y = # heads in n = 1000 tosses of a fair coin.
Im(Y ) = {0, 1, . . . , 1000}
Y is discrete.
(c.) W =time until a part on a machine fails.
∗ Im(W ) = (0, ∞)
∗ W is not discrete.
(d.) Z =trial on which the first head occurs
∗ Im(Z) = {1, 2, 3 . . .} = { positive integers}
∗ Z is discrete.
(e.) I = 1 if the amplifier fails during waranty period and 0 otherwise.
∗ Im(I) = {0, 1}
∗ I is discrete.
∗ I is a bernoulli random variable.
Random Variables and Bernoulli trials
Definition. A Bernoulli random variable is a r.v. with two possible outcomes. If X is a Bernoulli r.v., then Im(X) = {0, 1}, and
P (X = 1) = p,
P (X = 0) = q,
p + q = 1.
Random Variables and Independence
Definition: Two discrete r.v.’s X and Y are independent if
P (X = x, Y = y) = P (X = x)P (Y = y)
for all x ∈ Im(X) and y ∈ Im(Y ). In general, two r.v.’s X and Y are
independent if
P (X ∈ A, Y ∈ B) = P (X ∈ A)P (Y ∈ B)
for any two sets A ⊂ Im(X), and B ⊂ Im(Y ).
Note that P (X = x, Y = y) means P (X = x, and Y = y), and P (X ∈
A, Y ∈ B) means P (X ∈ A, and Y ∈ B).
Result: Two r.v’s X and Y are independent if
P (X ∈ A|Y ∈ B) = P (X ∈ A)
for all A ⊂ Im(X), and B ⊂ Im(Y ).
Intuitively, X and Y are independent if knowledge about Y tells us nothing
about what values are more or less likely for X.