Uniform Distribution

Uniform distribution (or) Rectangular Distribution

i. Uniform Distribution

A random variable X is said to have a continuous uniform distribution over an interval (a, b) if its probability density function is a constant = k, over the entire range of X.

(i.e.,) f(x) = K, a < x <b

= 0, otherwise

Since, the total probability is always unity.

a and b are said to be the two parameters of the uniform distribution on (a, b).

Note: The uniform distribution is also known as rectangular ni distribution, since y = f(x) describes a rectangle over the curve (the x-axis, between the ordinates x = a and x = b.

ii. The distribution function of the uniform distribution

The distribution function F (x) is given by

Since F(x) is not continuous at x = a and x = b, it is not differentiable at these points.

everywhere except at x = a and x = b.

Note: The pdf of a uniform variable 'X' in (-a, a) is given by

iii. Characteristic function of a uniform distribution

Characteristic function is given by

iv. Moments of a uniform distribution

Moments are given by

v. Mean deviation about the mean of uniform distribution

Mean distribution about mean is given by

Thus mean deviation of the uniform distribution

Example 1.10.1

Electric trains on a certain line run every half an hour between mid-night and six in the morning. What is the probability that a man entering the station at a random time during this period will have to wait atleast twenty minutes ? [A.U. A/M 2008]

Solution :

Let the random variable X denote the waiting time in minutes for the next train.

Given that a man arrives at the station at random

=> X is distributed uniformly on (0, 30) with density

Thus the probability that he has to wait for atleast 20 minutes is

Example 1.10.2

If the random variable X follows uniform distribution in (0, 1) with density f(x) = 1, 0 < x < 1 f(x) = 0, otherwise find the density function of -2 log X.

Solution:

Let Y = -2 log X. Thus the distribution function of Y is

Example 1.10.3

Show that for the uniform distribution :

 

the moment generating function about the origin is  Also, moments of even order are given by 

Solution :

Moment generating function about origin is given by

since there are no terms with odd powers of t in M_x(t) all moments of odd order about origin vanish.

(i.e.,) all moments of odd order about mean vanish. The moments of even order are given by

Example 1.10.4

If X is uniformly distributed over (0, 5), find the probability that (a) X < 2 (b) X > 3 (c) 2 < x < 5.

Solution:

Example 1.10.5

A random variable Y is defined as cos πx where X has a uniform p.d.f. over (-1/2, 1/2)

find mean and standard deviation. [A.U A/M 2017 R-13 RP]

Solution:

Example 1.10.6

If X is uniformly distributed over (-a, a), a > 0 find a, so as to satisfy the following: (a) P(X ≥ 1) = 1/3 (b) P(X > 1) = 1/2

Solution:

X is uniformly distributed over (-a, a)

Example 1.10.7

X is uniformly distributed with mean 1 and variance 4/3, find P(X < 0). [A.U Tvli M/J 2010, CBT A/M 2011]

Solution:

Given that, mean = 1 => 

Variance = 4/3 => 

Solving equations (1) and (2) we get, b = 3 and a = -1

Therefore, f (x) = 1/b-a = 1/4

Hence, P(X < 0) = 

Example 1.10.8

Buses arrive at a specified bus stop at 15 minutes intervals starting at 7 a.m. that is 7 a.m., 7.15 a.m., 7.30 a.m., etc. If a passenger arrives at the bus stop at a random time which is uniformly distributed between 7 and 7.30 a.m. find the probability that he waits (a) less than 5 minutes (b) atleast 12 minutes for a bus.

Solution :

Let X denotes the time that a passenger arrives between 7 and 7.30 a.m.

Then X ~ U(0, 30)

Then f(x) = 1/b-a = 1/30-0 = 1/30

(a) Passenger waits less than 5 minutes, (i.e.,) he arrives between 7.10 - 7.15 or 7.25 - 7.30

P(Waiting time less than 5 minutes)

= P(10 ≤ x ≤ 15) + P(25 ≤ x ≤ 30)

(b) Passenger waits atleast 12 minutes, (i.e.,) he arrives between 7 - 7.03 or 7.15 - 7.18.

P(Waiting time atleast 12 minutes)

= P(0 ≤ x ≤ 3) + P(15 ≤ x ≤ 18)

Example 1.10.9

A random variable 'X' has a uniform distribution over (-3, 3) compute (i) P(X < 2), P(|X| < 2), P(|x-2| < 2), (ii) Find K for which P(X > K) = 1/3 [A.U M/J 2009, CBT A/M 2011]

Solution :

We know that the p.d.f. of a random variable 'X' which is distributed uniformly in (-a, a) is

Example 1.10.10

If X is a random variable uniformly distributed in (0, 1), find the pdf of Y = sin x. Also find the mean and variance of Y. [A.U. M/J 2006]

Solution:

Given: Y = sin x

X has a uniform p.d.f over (0, 1)

g(y) = 1

Example 1.10.11

X is uniformly distributed random variable with mean 1 and variance 4/3. If 3 independent observations of X are made, what is the probability that all of them are negative. [A.U A/M 2015 (RP) R8]

Solution:

Let X ~ U (a, b)

and the p.d.f of X, which is uniformly distributed, is given by

.'. P(3 independent observations are negative)

= P (first, second and third observations is negative)

Example 1.10.12

The variates 'a' and 'b' are independently and uniformly distributed

in the interval (0, 6) and (0, 9) respectively. Find the probability that the equation x² - ax + b = 0 has two real roots.

Solution :

Condition for real roots of x² - ax + b = 0 is a² ≥ 4b

i.e., the condition is b ≤ a²/4

.'. Required probability = 

f(a, b) = f (a) f(b) = 1/54

The required probability 

Example 1.10.13

Starting at 5.00 a.m. every half hour there is a flight from San Francisco airport to Los Angeles International Airport. Suppose that none of these planes is completely sold out and that they always have room for passengers. A person who wants to fly to L.A. arrives at the airport at a random time between 8.45 a.m and 9.45 a.m. Find the probability that she waits (1) atmost 10 mins. (2) atleast 15 mins. [A.U M/J 2007]

Solution :

Let X be the uniform r.v. over the interval (0, 60)

Example 1.10.14

Show that for the uniform distribution f(x) = 1/2a -a < x < a the moment generating function about origin is sinh at/at [AU N/D 2006]

Solution:

Given : f(x) = 1/2a -a < x < a

To prove : M_X(t) = sinh at/at

Proof : We know that the moment generating function is

Example 1.10.15

If X is a random variable with a continuous distribution function F(X), prove that Y= F(X) has a uniform distribution in (0, 1). Further if

find the range of Y corresponding to the range of Y corresponding to the range 1.1 ≤ x ≤ 2.9 [A.U N/D 2010]

Solution :

The distribution function of Y

.'. the density function of Y is given by

.'. Y follows uniform distribution in (0, 1)

Given