a and b are the parameters of the Uniform distribution and we write X ~ U (a, b)

**Rectangular
or Uniform Distribution**

A random variable X is said to have a
continuous Uniform distribution over the interval (a, b) if its probability
density function is

*i. a *and* b *are the parameters of the Uniform
distribution and we write* X *~* U *(*a*,* b*)

ii.
The distribution is also known as Rectangular distribution, as the curve

*iii. y *=* f*(*x*) describes a rectangle over the
x-axis and between ordinates at x = a* *and *x*= *b*.

(iv)
*f* X ~ U (-a, a) then its p.d.f. is

If
*X* ~ *U* (200, 250) find its p.d.f and *P*
(*X* > 230)

If
*X* is a Uniform variate with the parameter 50 and100, find
the mean, median and standard deviation.

**Example 10.19**

If
*X* is a random variable having a uniform distribution *U*
(*a*,*b*) such that *P*(20<*X*<40)=0.2
and mean = 150, find a and b.

*Solution:*

(1)+(2)
implies 2b = 400 and b = 200.

Substituting
b in (2) we have a + 200 = 300 and that a =100.

a=100,
b=200

X~U(100,200)

**Example 10.20**

If
*X* is a Uniform variable U(*a*,*b*) with first and third quartiles
100 and 200, find the p.d.f of *X*.

*Solution:*

**Example 10.21**

Electric trains on a certain line run every 15 minutes 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 at least ten
minutes?

Let the random variable X denote the waiting time (in minutes).

The given assumption indicates that X is distributed Uniformly
on (0,15).

Tags : Definition, Formula, Characteristics, Solved Example Problems , 11th Statistics : Chapter 10 : Probability Distributions

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11th Statistics : Chapter 10 : Probability Distributions : Rectangular or Uniform Distribution | Definition, Formula, Characteristics, Solved Example Problems

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