## 3.6 Random Variables

### 3.6.1 Definition

A random variable (r.v.) is a transformation (function) of $$\Omega$$ into $$\mathbb{R}^n$$. This means that the results of random experiments will be transformed into numbers. Consider a random variable $$X$$. $$R_{X}$$ is the set of all possible values of $$X$$, called codomain. It can be thought of as a “numerical sample space” obtained from $$\Omega$$. A discrete random variable is such that $$R_{X}$$ is countable.

Example 3.22 (Discrete random variable) Consider the sample space defined in Example 3.13. Suppose you are interested in the random variable ‘sum of points’, denoted by $$S$$. The set of possible values of $$S$$ is $$R_{S} = \left\lbrace 2,3, \ldots, 12 \right\rbrace$$ and $$|R_{S}|=11$$. $$\\$$

Exercise 3.12 Consider the Example 3.13. Define the codomains of the following random variables:
a. $$D$$: ‘difference of points’.
b. $$P$$: ‘product of points’.
c. $$Q$$: ‘quotient of points’.

### 3.6.2Probability distribution

Let $$X$$ be a discrete random variable, where for each point of $$R_{X}$$ a probability (mass function) $$p(x) = Pr(X=x)$$ is associated, satisfying $$p(x) \ge 0$$ for all $$x$$ and $$\sum_{x \in R_{X}} p(x) = 1$$.

Example 3.23 (Probability with discrete r.v.) Suppose two consecutive tosses of a balanced coin. The sample space is $$\Omega = \left\lbrace HH,HT,TH,TT \right\rbrace$$, where $$H$$ represents ‘heads’ and $$T$$ ‘tails’. If we are interested in the random variable $$X$$: ‘number of heads’, the set of interest becomes $$R_{X} = \left\lbrace 0,1,2 \right\rbrace$$, where the element $$0$$ of the set $$R_X$$ is equivalent to event $$\left\lbrace TT \right\rbrace$$, $$1$$ to event $$\left\lbrace TH,HT \right\rbrace$$ and $$2$$ to $$\left\lbrace HH \right\rbrace$$. The probabilities are $p(0) = Pr(X=0)=Pr(\left\lbrace TT \right\rbrace) = \dfrac{1}{2} \times \dfrac{1}{2} =\dfrac{1}{4},$ $p(1) = Pr(X=1)=Pr(\left\lbrace TH,HT \right\rbrace)= \left( \dfrac{1}{2} \times \dfrac{1}{2} \right) + \left( \dfrac{1}{2} \times \dfrac{1}{2} \right) = \dfrac{2}{4} = \dfrac{1}{2},$ $p(2) = Pr(X=2)=Pr(\left\lbrace HH \right\rbrace)=\dfrac{1}{2} \times \dfrac{1}{2} =\dfrac{1}{4}.$ Note que $$Pr(X=0)+Pr(X=1)+Pr(X=2)=\dfrac{1}{4}+\dfrac{2}{4}+\dfrac{1}{4}=1$$. $$\\$$

Exercise 3.13 Get the probability distributions:
a. From the Example 3.22.
b. From item a. of Exercise 3.12.
c. From item b. of Exercise 3.12.
d. From item c. of Exercise 3.12.
$$\\$$

Exercise 3.14 Consider the Example 3.23.
a. Redo for three throws.
b. Redo for four throws.
c. Redo for $$n$$ throws.

### 3.6.3 Expected value

The expected value of a discrete random variable $$X$$ is given by

$$$E\left[ X \right] = \sum_{x} x \cdot p(x) \tag{3.44}$$$

the expected value of a function $$g(X)$$ is given by

$$$E\left[ g(X) \right] = \sum_{x} g(x) \cdot p(x) \tag{3.45}$$$

Example 3.24 (Expected value of discrete r.v. $$X$$ and $$X^2$$) From the Example 3.23 one can calculate $E(X) = 0 \times \dfrac{1}{4} + 1 \times \dfrac{2}{4} + 2 \times \dfrac{1}{4} = 1.$ This result was expected given the symmetry of the distribution. The expectation of $$g(X) = X^2$$ is given by $E(X^2) = 0^2 \times \dfrac{1}{4} + 1^2 \times \dfrac{2}{ 4} + 2^2 \times \dfrac{1}{4} = \dfrac{3}{2} = 1.5.$

### 3.6.4 Variance and standard deviation

The variance of a discrete random variable $$X$$ is given by

$$$V(X) = E( \left[ X - E(X) \right] ^2) = E(X^2) - \left[ E(X) \right] ^2 \tag{3.46}$$$

The standard deviation of a discrete random variable $$X$$ is given by

$$$D(X) = \sqrt{V(X)} \tag{3.47}$$$

Example 3.25 (Variance and standard deviation of a discrete r.v.) From the Example 3.24 it can be calculated $V(X) = \dfrac{3}{2} - 1^2 = \dfrac{1}{2} = 0.5$ $D(X) = \sqrt{0.5} \approx 0.7071$

Exercise 3.15 Calculate $$E(X)$$, $$E(X^2)$$, $$V(X)$$ e $$D(X)$$
a. In Exercise 3.13.
b. In Exercise 3.14.