Notes 3

Question: What is Stochastic Process?

Answer: A Stochastic Process is one that generates numbers with certain probability distribution (The word Stochastic simply means probablistic or randomly generated, but it is commonly used in case when referring to a sequence of results assumed to be generated by some underlying probability distribution.)

Question: What is Bernoulli trail?

Answer: A random expriment with exactly two results, success or failure.

Question: What is a Probabilty Mass Function?

Answer: Random variables have numeric range hence it is easy to do maths on them rather than doing them on events representing those variables. So we can define a Probability Mass Function of pmf for a random variable which gives out the probability that random variable have different values.

Question: What is Expectation?

Answer: The expectation is the mean of the random variable. So if $X$ is a random variable with pmf $p(x)$ such that $\sum_x|x|p(x) < \infty$, then expectation is

$$E(X) = \sum_xx p(x)$$

Example: If rolling one die and Y is the value on it face, what is the expectation?

Solution: We have total of 6 possibilties, hence $p(y) =\frac{1}{6}$

$$E(Y) = \sum_{1}^{6} y \frac{1}{6}$$

$$E(Y) = \frac{1}{6}\sum_{1}^{6} y$$

$$E(Y)= \frac{21}{6} = 3\frac{1}{2}$$

Question: What is Varience?

Answer: The varience of a random number is the measure of whether the values if the random variable tends to be consistent over trails or vairy a lot.

In terms of expectation, varience can be found by how much on an average the variable’s value deviate frim the variables expectation level.

$$Var(X) = E((X-E(X))^2)$$

$$Var(X)= E(X)^2 - E^2(X)$$

Question: What is Standard Deviation?

Answer: It is the square root of varience. It is denoted by $\sigma$

Example: What is Expectation and Varience of the sum of two unbaised dice rolls? $S = {2,3,…,12}$

Solution: Expectation is given in the formula above, let $E(X)$ be the Expectation of the sum of roll of two dices and $E(Y)$ be the Expectation of roll of one die$.

$$E(X) = E(Y) + E(Y)$$

which is calculate above,

$$E(X) = 3\frac{1}{2} +3\frac{1}{2} = 7$$

and Varience becomes

$$Var(X) = E((X-E(X))^2)$$

As we already know $E(X) = \sum_xx p(x)$ substituting it in above formula we get,

$$Var(X) = E((X -\sum_xp(x))^2)$$

$$Var(X) = \sum_x p(x)(x-\sum_xp(x)^2)$$

$$Var(X) = \frac{35}{6}$$