## Thursday, 7 August 2014

### chapter-18 Introduction to Random Variables

Suppose that a random experiment consists of tossing two dice, and the quantity of interest is the sum of the numbers on the two dice. Let us denote this sum by $X$.
$X$ will be termed a random variable of this experiment and it can take one of these possible values: $2,3,4,\ldots ,12$.Consider another random experiment wherein we toss a coin a $100$ times and we are interested in the number of Heads obtained, which is again a random variable of this experiment, with one of these possible values: $0,1,2,\ldots 100$
Of course, an experiment can have many random variables associated with it. For example, for the coin tossing experiment above, we can have so many possible random variables.
 ${X_1}$ : No. of tails ${X_2}$ : No. of Heads $-$ No. of Tails ${X_3}$ : The no. of tosses in the largest consecutive sequence of Heads $\vdots$ etc
Thus, we see that a random variable is a way of assigning to each outcome of the experiment, a single real number, which will vary with different outcomes of the experiment. So far, so good.
Now, we will try to understand what the probability distribution of a random variable means.
Consider once again the random experiment of rolling two dice and observing the sum of the numbers on the two dice, which we denoted by $X$$X$ can take a multitude of values in the following ways:
 $X$ Outcome(s) which gives this $X$ $2$ $(1,1)$ $3$ $(1,2)\,(2,1)$ $4$ $(1,3)\,(2,2)\,(3,1)$ $5$ $(1,4)\,(2,3)\,(3,2)\,(4,1)$ $6$ $(1,5)\,(2,4)\,(3,3)\,(4,2)\,(5,1)$ $7$ $(1,6)\,(2,5)\,(3,4)\,(4,3)\,(5,2)\,(6,1)$ $8$ $(2,6)\,(3,5)\,(4,4)\,(5,3)\,(6,2)$ $9$ $(3,6)\,(4,5)\,(5,4)\,(6,3)$ $10$ $(4,6)\,(5,5)\,(6,4)$ $11$ $(5,6)\,(6,5)$ $12$ $(6,6)$
Each value of $X$ has a certain probability of being obtained. For example
 $P(X = 5) = \dfrac{{{\rm{No}}\,{\rm{.}}\;{\rm{of\, outcomes \,for \,which\, }}X\,{\rm{ = 5}}}}{{{\rm{Total\, No\,}}{\rm{. of\, outcomes}}}}$ $= \dfrac{4}{{36}}$ $= \dfrac{1}{9}$
Let us plot the probabilities for each value of $X$:
This table gives us what is known as the probability distribution $(PD)$ of $X$, that is, it is a description of how the “probability is distributed” across different values of the random variable. In simple words, the $PD$ of any random variable$(RV) X$ tells us how probable each value of the $RV$ is.
The sum of the various probabilities in a $PD$ must be $1$, as should be obvious. This fact you are urged to confirm for the last table.
Let us write down another $PD$ as an example.