CHAPTER 5: Some Discrete Probability Distributions
Discrete Uniform Distribution: 5.2
Defnition:
If the random variable
X
assumes the values
x
1
, x
2
, . . . , x
k
with equal probabilities, then the
discrete uniform distribution is given by
f
(
x
;
k
) =
1
k
,
x
=
x
1
, x
2
, . . . , x
k
Example:
When a die is tossed, each element of the sample space
S
=
{
1
,
2
,
3
,
4
,
5
,
6
}
occurs with probability
1/6. Therefore, we have a uniform distribution, with
f
(
x
; 6) =
1
6
,
x
= 1
,
2
,
3
,
4
,
5
,
6
.
μ
=
1 + 2 +
···
+ 6
6
= 3
.
5
σ
2
X
=
35
12
Binomial Distribution: 5.3
NOTE:
Multinomial Distribution is not required.
The Bernoulli Process
•
The experiment consists of
n
repeated trials.
•
Each trial results is an outcome that may be classiFed as a success or a failure.
(i.e.
heads/tails,
correct/erroneous bits, good/defective items, active/silent speakers).
•
The probability of success denoted by
p
, remains constant from trial to trial.
•
The repeated trials are independent.
The number
X
of successes in
n
Bernoulli trials is called a
binomial random variable
. ±or example
X
could be the number of heads in
n
tosses of a coin.
Example:
An early warning detection system for aircraft consists of four identical radar units operating
independently
of one another.
Suppose that each has a probability of 0
.
95 of detecting an intruding
aircraft. When an intruding aircraft enters the scene, let
X
= the number of radar units that do not detect the plane.
•
•
•
•
1
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View Full DocumentThe probability distribution of a binomial random variable:
Example
For the last example
P
(
X
= 3) =
P
(SSSF, SSFS, SFSS, FSSS) =.
The probability distribution of a binomial random variable is called a
binomial distribution
, and its values
will be denoted by
b
(
x
;
n, p
).
b
(
x
;
n, p
) =
±
n
x
²
p
x
q
n

x
,
x
= 0
,
1
, . . . , n
where
q
= 1

p
.
Example:
Find the probability that seven of 10 persons will recover from a tropical disease if we can assume
independence and the probability is 0
.
80 that any one of them will recover from the disease.
Solution:
Example:
Experience has shown that 30% of all persons a±icted by a certain illness recover.
A drug
company has developed a new medication. Ten people with the illness were selected at random and injected
with the medication; nine recover shortly thereafter. Suppose that the medication was absolutely worthless.
What is the probability that at least nine of ten injected with the medication will recover?
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 Spring '09
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