Download Section 4: Random Variables and Probability

Section 4: Random Variables and
Probability Distributions, Independent
Random Variables
September 11th, 2014
Lesson 6
A random variable is a function X on a probability space S
whose output is a real number X (s). In general, we think of a
random variable as associating some number and probability
to the outcome of a given experiment.
Suppose we take a gamble involving flipping a fair coin. If
heads is flipped, $1 is paid out. If tails is flipped, $2 is paid
out. The random variable X that describes this experiment
would take the values 1 and 2, the outcomes of the
experiment, and the associated probabilities would be 12 for
each outcome.
Note that the specific experiment doesn’t really matter. A
gamble where rolling an even number on a fair die pays out $1
and rolling an odd number pays out $2 would have the same
random variable.
Lesson 6
A random variable is discrete and has a discrete
distribution if the values it takes come from a finite or
countably infinite sequence (i.e. a subset of the integers).
Suppose we flip a coin. Let X = 1 if the first head occurs on
an even-numbered flip. Let X = 0 if the first head occurs on
an odd-numbered flip. Let Y = n be the number of the toss
on which the first head occurs.
Then X and Y are both discrete random variables, with X
taking values in the set {0, 1} and Y taking values in the set
{1, 2, 3, . . .}
Lesson 6
The probability function (pf) of a discrete random variable will
typically be denoted by f (x ) or p(x ), and it is equal to the
probability that the value x occurs for the random variable X .
We often denote this probability by P[X = x ]. A pf must
satisfy the conditions
(i) 0 ≤ p(x ) ≤ 1 for all x , and
X
(ii)
p(x ) = 1.
x
We also have
P[X ∈ A] =
X
p(x ) = P[A].
x ∈A
Lesson 6
Consider the experiment of rolling a fair six-sided die, with
probability space S = {1, 2, 3, 4, 5, 6}. Let X be the random
variable describing this experiment. Note that X is discrete.
For any s ∈ S, we have
1
f (s) = P[X = s] = .
6
Let A be the event “rolls an even number”, and let B be the
event “rolls a number that doesn’t start with a ’t’ or an ’f’.
Then
1
P[X ∈ A] = P[X = 2] + P[X = 4] + P[X = 6] = ,
2
and
1
P[X ∈ B] = P[X = 1] + P[X = 6] = .
3
Lesson 6
A continuous random variable takes values in some interval
of real numbers.
Given a continuous random variable X , a probability density
function (pdf) for X is a function f (x ) which is continuous at
all but finitely many points. To find the probability that X
takes a value in some interval, we integrate f (x ) over that
integral. That is,
P[X ∈ (a, b)] = P[a < X < b] =
Z b
a
Lesson 6
f (x ) dx .
Note that P[X = a] = aa f (x ) dx = 0. This means that the
probability that X takes on any one value is 0. Thus
R
P[a < X < b] = P[a ≤ X < b]
= P[a < X ≤ b]
= P[a ≤ X ≤ b]
If f (x ) is a pdf, then it must satisfy the conditions
(i) f (x ) ≥ 0 for all x , and
(ii)
Z ∞
f (x ) dx = 1.
−∞
Lesson 6
Example (4.1)
Suppose that X has density function
(
f (x ) =
3x 2
0
0<x <1
elsewhere
(a) Show that f satisfies the conditions of a pdf
(b) Calculate P[.3 < X ≤ .8]
(c) Calculate P[X ≥ .5]
Lesson 6
For a random variable X , the cumulative distribution
function (cdf) of X is the function
F (x ) = P[X ≤ x ].
It is the cumulative probability to the left of (and including) x .
The survival function is the complement
S(x ) = 1 − F (x ) = P[X > x ].
Lesson 6
If X is a discrete random variable, then F (x ) =
and then F (x ) is a step function.
P
w ≤x
p(w ),
x
If X is a continuous random variable, then F (x ) = −∞
f (t) dt.
By the Fundamental Theorem of Calculus, we have
R
F 0 (x ) =
d
dx
Rx
−∞
f (t) dt = f (x ).
For any cdf,
P[a < X ≤ b] = F (b) − F (a),
lim F (x ) = 1, and
x →∞
lim F (x ) = 0.
x →−∞
Lesson 6
The condition that a collection of random variables are
independent is exactly what one would expect.
If the random variables X and Y are independent, then we
have
P[(a < X ≤ b) ∩ (c < Y ≤ d)] = P[a < X ≤ b] · P[c < Y ≤ d].
Example (4.2)
An ordinary single die is tossed repeatedly and independently
until the first even number turns up. The random variable X is
defined to be the number of the toss on which the first even
number turns up. Find the probability that X is an even
number.
Lesson 6
Example (4.3)
Let X be a continuous random variable with density function
(
f (x ) =
6x (1 − x )
0
Calculate P[|X − 38 | > 81 ].
Lesson 6
0<x <1
otherwise
Example (4.4)
The lifetime of a machine part has a continuous distribution
on the interval (0, 40) with probability density fuction f , where
f (x ) is proportional to (x + 7)−2 . Calculate the probability
that the lifetime of the machine part is less than 5.
Lesson 6