## Summary

- In uniform distribution the random variable is a continuous random variable
- The probability density function is calculated as:
- Mean
- Variance
- The cumulative distribution function is calculated by integrating the probability density function f(x) to give
- Standard deviation is the under root of variance

In uniform distribution you should know that random variable is a continuous random variable. In continuous uniform distribution it takes infinite number of real values in an interval.

For example, if we say that it is observed in a school, over a period of 2 months that the teacher arrives in school earliest by 4 minutes, before school starts or latest by 6 minutes after school starts.

*(note: we are assuming that school starts at sharp 8 in the morning)*

Thus looking at the figure below, we can deduce that the probability of a teacher entering class anywhere between 7:56 to 8:06 is constant.

However, in order to find probability density function, we can say that, we know that the since, -4 till 6 are the only possibilities, we can thus say that the probability is greater than 0 only if the interval of x is between -4 and 6 .

Otherwise we can say that it is equal to 0 ( meaning, the teacher has taken a leave).

We can thus write the uniform distribution as:

X ~ U(A, B)

Which in this case will be:

X ~ U(-4, 6)

Now, in order to find the interval between A and B we use the probability density function. Which is calculated as:

In addition, if you are asked in a question, to find the mean we use the following formula:

Mean

But, we also need to understand how to derive this formula:

Variance

#### Cumulative Distribution Function

The cumulative distribution function of a continuous random variable, is known give the probabilities and is calculated by integrating the probability density function f(x) between the limits and x.

#### Example #1

Q. A random variable is uniformly distributed over the interval 2 to 10. Find its variance.

*Solution:*

We have been given the interval 2 to 10 so we know, a = 2 and b = 10

The formula for variance is .

Plug, in the values to calculate the answer:

*Ans*

#### Example #2

Q. A zero mean random signal is uniformly distributed between limits -a and +a and its mean square value is equal to its variance. Then the standard deviation of the signal is?

*Solution:*

We know that standard deviation is the under root of variance thus:

Next, the values of a and b are given to us:

a= -a and b = +a

Now plug the values into the formula above to get

**Ans**

### Example#3

Q. Suppose a train is delayed by approximately 60 minutes. What is the probability that train will reach by 57 minutes to 60 minutes?

*Solution:*

Interval of probability distribution a = 0 minutes b = 60 minutes

We need to find the probability density function with the given values of a and b between the probability :

We can thus, say the probability that train will reach by 57 minutes to 60 minutes is .