CHAPTER 3: Random Variables and Probability Distributions
CHAPTER 3: Random Variables and Probability Distributions
Concept of a Random Variable: 3.1
? The outcome of a random experiment need not be a number.
? However, we are usually interested not in the outcome itself, but rather in some measurement of the
outcome.
Example: Consider the experiment in which batteries coming off an assembly line were examined until a
good one (S) was obtained.
S = {S, FS, FFS, . . .}.
We may be interested in the number of batteries examined before the experiment terminates.
A random variable is a function that associate a real number with each element in the sample space.
Example: Tossing two coins
S = {HH, TT, HT, TH}
Let X = # of heads observed.
Example: A group of 4 components is known to contain 2 defectives. An inspector tests the components
one at the time until the 2 defectives are located. Let X denote the number of the test on which the second
defective is found.
Two types of random variables
? A discrete random variable is a random variable whose possible values either constitute a finite set
or else can be listed in an infinite sequence.
? A random variable is continuous if its set of possible values consists of an entire interval on the
number line.
Many random variables, such as weight of an item, length of life of a motor etc., can assume any value
in certain intervals.
1
Discrete Probability Distributions: 3.2
Probability mass function of a discrete random variable X is defined by
f (x) = P (X = x)
Example: tossing two coins
X = # of heads.
f (0) = P (X = 0) = P (TT) = 1/4
f (1) = P (X = 1) = P (HT, TH) = 1/2
f (2) = P (X = 2) = P (HH) = 1/4
Example: An information source produces symbols at random from a five-letter alphabet:
S = {a, b, c, d, e}.
The probabilities of the symbols are
p(a) =
1
1
1
1
, p(b) = , p(c) = , p(d) = p(e) =
.
2
4
8
16
A data compression system encodes the letters into binary strings as follows:
a
b
c
d
e
1
01
001
0001
0000
Let the random variable Y be equal to the length of the binary string output by the system.
f (1) = P (Y = 1) =
f (2) = p(Y = 2) =
f (3) = p(Y = 3) =
f (4) = p(Y = 4) =
f (x) = P (X = x) satisfies the following conditions:
1. fP(x) ¡Ý 0
2.
f (x) = 1
Example: A box contains 5 balls numbered 1, 2, 3, 4, and 5. Three balls are drawn at random and without
replacement from the box. If X is the median of the numbers on the 3 chosen balls, then what is the
probability function for X, where nonzero?
Solution
2
Example: Determine c so that the function f (x) can serve as the probability mass function of a random
variable X:
f (x) = cx for x = 1, 2, 3, 4, 5
Solution:
The cumulative distribution function: F (x) of a discrete random variable X with probability mass function
f (x) is defined for every number x by
X
f (t)
F (x) = P (X ¡Ü x) =
t¡Üx
Example: Assume that
f (2) = p(X = 2) = 1/6 f (3) = p(X = 3) = 1/3
f (4) = p(X = 4) = 1/2
Then
F (2) =
F (3) =
F (4) =
F (x) =
Example: A mail order computer business has six telephone lines. Let X denote the number of lines in use
at a specified time. Suppose that the probability mass function of X is given below
x
p(x)
0
0.10
1
0.15
2
0.20
3
0.25
4
0.20
5
0.06
6
0.04
a. Find the cumulative distribution function
b. Find the probability that
{at most 3 lines are in use}.
c. Find the probability that
{at least 4 lines are in use}.
3
Example:
If X has the cumulative distribution function:
find the probability mass function.
?
0
?
?
?
?
? 1/3
1/2
F (x) =
?
?
? 5/6
?
?
1
if
if
if
if
if
x ................
................
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