1-The Uniform random variable 

A random variable is said to be uniformly distributed over the interval

(0, 1) if its probability density function is given by 

𝑓(𝑥)    is  a  density  function,  since 𝑓(𝑥)≥0  and 

 Because 𝑓(𝑥)>0  only when x ∈ (0, 1), it follows that X must assume a value in interval (0, 1).

Also, since f (x) is constant for x ∈ (0, 1), X is just as likely to be near any value in (0, 1) as it is to be near any other value.

To verify this statement,

for any 0 < a < b < 1, 

In other words, the probability that X is in any particular subinterval of (0, 1) equals the length of that subinterval.

we say that X is a uniform random variable on the interval (𝛼,𝛽)if the probability density function of X is given by 

Let X be uniformly distributed over (α, β). Find (a) E(X) and (b) Var(X). 

(a) 

In words, the expected value of a random variable that is uniformly distributed over some interval

is equal to the midpoint of that interval. (b) To find Var(X), we first calculate 

Therefore,  the  variance  of  a  random  variable  that  is  uniformly distributed

over some interval is the square of the length of that interval divided by 

8:If X is uniformly distributed over (0, 10), calculate the probability that (a) X < 3, (b) X > 6, and (c) 3 < X < 8.

(a) 

Suppose  the  amount  of  gasoline  sold  daily  at  a  service  station  is uniformly distributed with a

minimum of 2,000 gallons and a maximum of 5,000 gallons.

(a) What is the probability that daily sales will fall

between 2,500 gallons and 3,000 gallons? 

(b) What is the probability that the service station will sell at least 4,000 gallons? 

(c) What is the probability that the service station will sell exactly 2,500 gallons?