1. Function of a random variable Let X be a random variable in a probabilistic space with a probability distribution F(x) Sometimes we may be interested in another random variable Y that is a function of X, that is, Y = g(X). The question is whether we can establish the probability distribution G(y) of Y.

2. Let g(x) be increasing on R(X). Since F(x) is the probability distribution of X and G(y) of Y, we can write and g(x) is increasing and so it has an inverse This means that

3. If g(x) is decreasing on R(X), we can proceed in a similar way, but since the inverse of a decreasing function is again decreasing, the inequality will revert the relation sign and so so that

4. Random variable X has the standardized normal distribution, calculate the distribution of the random variable Y = X 2. Sometimes we can use even more sophisticated methods as shown in the following example.

5. We cannot calculate the last integral exactly, but we can estab- lish the probability density of Y by differentiating G(y)

6. This is the probability density of the random variable X 2. We have, in fact, calculated the density of the chi-squared distribution with one degree of freedom.

7. Example Random variable X has a uniform distribution on [0,1]. Find a transformation g(X) such that Y = g(X) has a distribution F(y). X has the density and the distribution 1 1 x F(x)F(x)

8. F(x) is a distribution and as such has R as the domain and [0,1] as the range. This means that, if F(x) is increasing, it has an inverse F -1 (x) that is also increasing with range R and domain [0,1]. Consider the transformation Y = F -1 (x). We have Thus, we have shown that F(y) is the distribution of Y = F -1 (x). This can be used for example for simulating a distribution using a pocket calculator.

9. Random vector Height: 115 cm Weight: 17 kg No of children: 0 Employed: No Height: 195 cm Weight: 98 kg No of children: 4 Employed: Yes Height: 170 cm Weight: 80 kg No of children: 2 Employed: No POPULATION Persons chosen at random

10. The sequence of random variables X 1 = Height X 2 = Weight X 3 = Number of children X 4 = Employed is an example of a random vector. Generally, a random vector assigns a vector of real numbers to each outcome n times

11. Given a probabilistic space, a mapping is called a random vector if for every

12. Probability distribution of a random vector Let us consider a probabilistic space for a random vector we define its probability distribution as follows

13. Properties of the distribution of a random vector The probability distribution of a random vector has the following properties is increasing and continuous on the left in each of its independent variables

14. Discrete random vectors A random vectoris called discrete if its range is a finite or a countable set of real vectors For a discrete random vector, we can define the probability function

15. The relationship between a probability distribution and probability function

16. Continuous random vectors A random vectoris called continuous if its range includes a Cartesian product of n intervals If a functionexists such that we say thatis the probability density of the random vector

17. Marginal distributions For a random vector with a distribution we define marginal distributions is a limit ofwith all variables except x i tending to infinity

18. If a random vector is discrete, we define its marginal probability functions where the summation is done over all the values of the variables

19. If a random vector is continuous, we define its marginal probability densities

20. By considering a marginal distribution of a random vector we actually define a single random variable X i by "neglecting" all other variables.

21. Example for n = 2 Let (X,Y) be a discrete random variable with X taking on values from the set {1,2,3,4}, Y from the set {-1,1,3,5,7} and the probability function given by the below table. Calculate the marginal probability functions of the random variables X and Y.

23. Let be a random vector with a distribution If we say that and marginal distributions are independent random variables. Ifare independent and have a probability function p or density f, it can be proved that also or

24. Correlation coefficient of two random variables Let us consider a random vector (X,Y) with a distribution F(x,y). Using the marginal distributions F x (x,y) and F y (x,y) we can define the expectancies E(X) and E(Y) and variances D(X) and D(Y). We define the covariance of the random vector (X,Y) cov(X,Y) = E(XY) – E(X)E(Y) or depending on whether (X,Y) is discrete or continuous. Then is the probability function or density. with

25. The correlation coefficient is then defined as has the following properties

26. Example Calculate the correlation coefficient of the discrete random vector (X,Y) with a probability function given by the below table

28. Note that, generally, it is not true that implies that X and Y are independent as proved by the following example The correlation coefficient is zero as can be easily calculated, but we have, for example, so that X and Y cannot be independent.