17 - Random variables and expected values

# 17 - Random variables and expected values ## Random variables > A random variable is neither random nor a variable Definition: Let $S$ be a sample space. Then a **random variable** on $S$ is a function $X:S \rightarrow \mathbb{R}$ Where $S$ is an outcome of an experiment mapped to a real number. Ex: Rolling two dice $S = \{(i, j) : i,j \in \{1,2,3,4,5,6\}\}$ $X(i,j) = i + j$ Ex: Flipping three coins $S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$ Random variables you could define: $X(\omega ) = \text{The number of heads in } \omega$ $X(\omega ) = 0 \text{ if } \omega= TTT$ $X(\omega ) = 1 \text{ if } \omega \in \{HTT, THT, TTH\}$ $X(\omega ) = 2 \text{ if } \omega \in \{HHT, HTH, THH\}$ $X(\omega ) = 3 \text{ if } \omega= HHH$ $Y(\omega ) = 1 \text{ if outcome is all heads or all tails}$ $Y(\omega ) = 0 \text{ otherwise}$ ### Defining events in terms of random variables An informal way of describing the event in which the outcome results in a value of something Ex: $X=0 = \{TTT\}$ $X=1 = \{HTT, THT, TTH\}$ $"X = \chi" = \{\omega \in S: X(\omega)=\chi \}$ "$X$ equals some value is another way of saying: for every outcome $\omega \in S$ $X$ of $\omega$ equals that value" ### Random independence Definition: In a probability space $(S, Pr)$ two random variables $X$ and $Y$ are independent if for every $x,y \in \mathbb{R}$ $Pr(X=x \cap Y=y) = Pr(X=x) \times Pr(Y=y)$ The coin toss example is an example of non independence $Pr(X=1 \cap Y=1) = Pr(\emptyset) \neq \frac{3}{8} \times \frac{2}{8}$ Another ex: $Z(\omega ) = 1 \text{ if the first coin is head} = \{HHH, HHT, HTH, HTT\}$ $Z(\omega ) = 0 \text{ if the first coin is tail} = \{THH, THT, TTH, TTT\}$ Are $Y$ and $Z$ independent? $Pr(Y=0) = \frac{6}{8}$ $Pr(Y=1) = \frac{2}{8}$ $Pr(Z=0) = \frac{1}{2}$ $Pr(Z=1) = \frac{1}{2}$ $Pr(Y=0 \cap Z=0) = \{THH, THT, TTH\} = \frac{3}{8} = \frac{6}{8} \times \frac{1}{2}$ $Pr(Y=0 \cap Z=1) = \{HHT, HTH, HTT\} = \frac{3}{8} = \frac{6}{8} \times \frac{1}{2}$ $Pr(Y=1 \cap Z=0) = ...$ We can deduce using the lemma that if $A \cap B$ are independent then $A \cap \bar{B}$ are also independent to show that the rest are independent They're independent. ~~Source: trust me bro~~ Definition: A sequence of random variables $X_1,...,X_m$ are pairwise/mutually independent if for all $X_1,...,X_m \in \mathbb{R}$ the events $X_1=x_1, X_2=x_2,..., X_m=x_m$ are pairwise/mutually independent. ## Expected values The expected value of a random variable $X$ is $E(X) = \sum_{\omega \in S} Pr(\omega ) \times X(\omega ) = \sum_{x \in \mathbb{R}} Pr(X=x)\times x$ The sum of all of the random variables assigned to each outcome multiplied by the value of the probability of that outcome. Also known as the mean, expectation, ~~average~~ Example: Roll a die $S=\{1,2,3,4,5,6\} Pr(\omega ) = \frac{1}{6}$ Random variable: $X(i)=i$ The random variable $X$ is equal to the result of the dice roll $E(X) = \sum_{\omega \in S} Pr(\omega) \times X(\omega) = \frac{1}{6}\sum_{i=1}^6 i = \frac{1}{6} \times \frac{6 * 7}{2} = \frac{7}{2} = 3.5$ ## Something difficult and painful Rolling two die $X(i,j) = i + j, Pr(\omega ) = \frac{1}{36}$ What is the expected value of rolling two die under this definition of $X$? $E(X) = \sum_{\omega \in S} Pr(\omega) \times X(\omega)$ $= \sum_{i=1}^6 \sum_{j=1}^6 Pr(i,j) \times X(i,j) = \frac{1}{36} \times \sum_{i=1}^6 \sum_{j=1}^6 (i+j)$ (probability of outcome rolling is always the same) $= \frac{1}{36} \sum_{i=1}^6(6i + \sum_{j=1}^6 j) = \frac{1}{36} \sum_{i=1}^6 (6i + \frac{6 * 7}{2})$ $= \frac{1}{36}(\frac{6 * 6 * 7}{2} + \frac{6 * 6 * 7}{2}) = \frac{7}{2} + \frac{7}{2} = 7$ Using the other definition of calculating expected values $E(X) = \sum_{x \in \mathbb{R}} Pr(X=x) \times x$ $= \frac{1}{36} \times 2 + \frac{2}{36} \times 3 + \frac{3}{36} \times 4 + \frac{4}{36} \times 5 + \frac{5}{36} \times 6 + \frac{6}{36} \times 7 + \frac{5}{36} \times 8 + \frac{4}{36} \times 9$ + $\frac{3}{36} \times 10 + \frac{2}{36} \times 11 + \frac{1}{36} \times 12 = 7$ ## The linearity of expectation Lemma (Linearity of expectation): For ***any*** two random variables $X$ and $Y$ $E(X + Y) = E(X) + E(Y)$ $"X+Y" = \text{A random variable } Z(\omega ) = X(\omega ) + Y(\omega )$ Ex: $A(i, j) = i$ $B(i, j) = j$ $\tilde{X}(i,j) = i+j = A(i,j) + B(i,j)$ Like saying $\tilde{X} = A + B$ $E(\tilde{X}) = E(A+B) = E(A) + E(B)$ $E(A) = \sum_{i=1}^6\sum_{j=1}^6 Pr(i,j) \times A(i,j) = \frac{1}{36}\sum_{i=1}^6\sum_{j=1}^6 i$ $= \frac{1}{36} \frac{6 * 6 * 7}{2} = \frac{7}{2}$ $E(B) = ... \frac{7}{2}$ ###### tags: `COMP2804` `Probability`