Importance of Statistics and Introduction to Random Variables:

The field of statistics deals with the collection of data, presentation, analysis and use of data to make the decisions and draw valid conclusions in the presence of variability, solving problems and designing the products. Statistics can be described as the study of how to make inferences and decisions in the case of uncertainty and variability .

The role of statistics is indispensable in Engineering. All the practicing engineers soon after they enter into work place, they realize the importance of statistics starting from designing the product to making the product to work.

Variability is omnipresent (being everywhere) in the business world .To model variability, polarizability, we need the concept of Random variable. A Random variable is a numerically valued variable which takes off different values with given probabilities.

Ex:

• The return of investment in a period of 1st year.
• Number of customers entering into a store in 1 hour etc.

Random Variable :

A real number X is associated with the out come of a random experiment is called as Random variable.

(or)

A real function X when each sample point in ‘S’ is mapped onto a real number , then it is called a Random Variable .

(or)

The quantities which vary with some probability are called as Random Variables .

Ex :

Suppose two fair coins are tossed . The we know that sample space , S = {HH,TT,HT,TH} .

Let ‘X’ be a random variable denoting heads .

Then X = 1 $$\Rightarrow$$ getting 1 head.

Then p(X = 1) =$$ \frac{2}{4} = \frac{1}{2}$$

P(X =2 ) = $$\frac{1}{4}$$

P(X=0) =$$ \frac{1}{4}$$

X =x =0,1,2 P(X=x) $$\frac{1}{4} ,\frac{1}{2},\frac{1}{4}$$

Note :

If x,y are random variables and ‘a’ and ‘b’ are constants , then x + y , xy , x+a ,ax+b , ax+by_ _ _ _ _ are all random variables.

Cumulative Distributive function :

If X is a random variable taking the values x, ie -$$\infty < X <\infty $$, then the function F(x) , i.e., F(x) = p(X$$\leq $$x ) is called as cumulative distribution function . It is written as Cumulative distribution function .

It satisfies the following properties :

• P(a$$\leq$$ x <b) = F(b) – F(a)
• F(x) $$\geq 0$$ and $$F(x) \leq 1 (or ) 0 \leq F(x) \leq 1$$
• If x < Y , then F(x) <F(y).
• F( -$$\infty) = 0$$ and $$F(\infty) =1 .$$

For continuous distribution P(a<x<b) = P$$(a\leq x \leq b) = p(a<x\leq b) = p(a\leq x \leq b)$$