Uniform Distribution Formula

Theoretical Mean Formula

\(\mu\) = \(\frac{x+y}{2}\)

Standard Deviation Formula

\(\sigma = \sqrt{ \frac{(y-x)^{2}}{12}}\)

Example of Uniform Distribution

Example: The data in the table below are 55 times a baby yawns, in seconds, of a 9-week-old baby girl.

10.4

19.6

18.8

13.9

17.8

16.8

21.6

17.9

12.5

11.1

4.9

12.8

14.0

22.8

20.8

15.9

16.3

13.4

17.1

14.5

19.0

22.8

1.3

0.7

8.9

11.9

10.9

7.3

5.9

3.7

17.9

19.2

9.8

5.8

6.9

2.6

5.8

21.7

11.8

3.4

2.1

4.5

6.3

10.7

8.9

9.7

9.1

7.7

10.1

3.5

6.9

7.8

11.6

13.8

18.6

  • The sample mean = 11.49
  • The sample standard deviation = 6.23.

As assumed, the yawn times in secs, it follows a uniform distribution between 0 to 23 seconds(Inclusive).

So, it is equally likely that any yawning time is from 0 to 23.

In this example,

The theoretical mean = \(\mu\) = \(\frac{x+y}{2}\) \(\mu\) = \(\frac{0+23}{2}\) = 11.50

standard deviation = \(\sigma = \sqrt{ \frac{(y-x)^{2}}{12}}\)

standard deviation = \(\sqrt{\frac{(23-0)2}{12}}\) = 6.64 seconds.


Practise This Question

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type
[f(x)h(x)+g(x)]dx
Let f(x)h(x)dx=I1 and g(x)dx=I2
Integrating I1 by parts, we get
I1=f(x)h(x)dx{f(x)h(x)dx}dx
Now find (1log x1(log x)2)dx (x > 0)