Tutorial 6

1. Tutorial 6 Mode and Median of Random Variable, Gamma Distribution

2. Mode of random variable • Discrete R.V.: If T = k is the mode, then, • Continuous R.V.: If a is the mode, then • Position of the maximum in p.d.f / p.m.f

3. Median of random variable • Discrete R.V.:If T = i is the median, then, • Continuous R.V.: If a is the median, then

4. Example - Binomial R.V • X is a random variable with parameter (n,p), find the median and mode of X? • Median: a minimum k such that

5. Example - Binomial R.V • Mode: a maximum k such that

6. Example – Exponential R.V • X is a random variable with p.d.f, • Find its mode and median. • Find its generating function, and thereby its skewness and kurtosis.

7. Example – Exponential R.V • Mode: As f(x) is strictly decreasing function, its maximum occurs at x=0.  0 is the mode. • Median:

8. Example – Exponential R.V • Moment Generating Function: • By finding E[X], E[X²], E[X³],…from M(t), skewness and kurtosis can be found.

9. Gamma Distribution • A random variable is said to have a gamma distribution with parameters (t, ), if its density function is given by

10. Gamma Distribution

11. Gamma Distribution • The cumulative density function,

12. Mean of Gamma Distribution • Consider X as a gamma random variable with parameter t and .

13. Mean of Gamma Distribution • Mean of Gamma(t , ) = t/ 

14. Variance of Gamma Distribution • First calculate E[X2]:

15. Variance of Gamma Distribution • Variance ofGamma(t , ) = t/ ²

16. Mode of Gamma Distribution • Since • Mode of Gamma(t, ) = (t-1)/

17. HW3 Q10 Hints. • Given the mean and mode. • i) First find the value of t and ’s. • ii) Once you find the t, , you can plot the income distributions.

18. Income Distribution >> x=[0:60000]; >> t=4; >> m=1/3000; >> y=m*exp(-1*m*x).*(m*x).^(t-1)/gamma(t); >> plot(y)

19. Income Distribution

20. Gini Index