Uniform Probability Distributions
Definition:
A random variable 𝑌 has a uniform distribution on ( 𝜃1 , 𝜃2 ) if and only if
the pdf of Y is given by
1
, 𝜃1 ≤ 𝑦 ≤ 𝜃2
𝑓(𝑦) = {𝜃2 − 𝜃1
0,
elsewhere
The quantities 𝜃1 and 𝜃2 are parameters of the uniform density function.
Example, If 𝜃1 = 0, and 𝜃2 = 10, then 𝑓(𝑦)is
Example 4.7
Arrivals of customers at a checkout counter follow a uniform distribution.
It is known that, during a given 30-minute period, one customer arrived at
the counter. Find the probability that the customer arrived during the last
5 minutes of the 30 minute period. (Answer is 1/6)
Theorem:
If random variable 𝑌 has a uniform distribution on interval (𝜃1 , 𝜃2 ), then
𝜃1 + 𝜃2
𝜇 = 𝐸(𝑌) =
2
(𝜃2 − 𝜃1 )2
𝜎 = 𝑉(𝑌) =
12
Prove the theorem yourself.
2
1
Exercise 4.45
Upon studying low bids for shipping contracts, a microcomputer
manufacturing company finds that intrastate contracts have low bids that
are uniform distributed between $20,000 and $25,000. Find the probability
that the low bid on the next intrastate shipping contract
a. Is below $22,000.
b. Is in excess of $24,000.
c. Find 𝐸(𝑌),the expected value of low bids.
Solve:
Convert units to thousands of dollars. Then 𝜃1 = 20, and 𝜃2 = 25.
1
1
=
𝑓(𝑦) = {25 − 20 5 , 20 ≤ 𝑦 ≤ 25
0,
otherwise
22 1
1
22
5
5
20
25 1
1
25
5
5
24
a. 𝑃(𝑌 < 22) = ∫20 𝑑𝑡 = 𝑡]
b. 𝑃(𝑌 > 24) = ∫24 𝑑𝑡 = 𝑡]
c. 𝐸(𝑌) =
20+25
2
= 22.5
2
1
2
5
5
1
1
5
5
= (22 − 20) = = 0.4
= (25 − 24) = = 0.2