RANDOM VARIABLE

INTRODUCTION:

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And Some think it is very tough.

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✦ A random variable is often described as a variable whose values are determines  by outcomes of a random experiment.

OR, 

      We assigned some real values to each of the outcomes of a sample space ,which is called the random variable.

✦ A random variable is a function , X :S ➝ R i.e a random variable is a function whose domain is the sample space of a random experiment(S) and whose range is a real line(R).

 Example: Let X is a random variable , the no. of heads in the experiment of tossing a coin twice , then in this case  S ={ HH,HT,TH,TT} and X ={2,1,1,0} i.e X(HH)=2, X(HT)=1, X(TH)=1,X(TT)=0. Thus the domain of  X is S and range is {0,1,2}.

✦ For a mathematical and rigorous definition of the random variable , let us consider the triplet (S,B,P), where S is the sample space B is the 𝞼-field of subsets in S and P is the probability function on B.

   So mathematically , a random variable is a function X(𝝎) with domain S and range (-∞ , ∞) ,such that for every real number 'a' , the event [ 𝝎 :  X(𝝎) ≤ a ] ∈ B.

✦ Random variable X can be written as P( X ≤ a ) to make probability statements about 'a'.

     For simple example given above , we should write P( X ≤ 1 ) = P{ HH,HT,TH,TT } = 3/4.

NOTATION:

✦ One dimensional random variable will be denoted by capital letters X,Y,Z,... etc. A typical outcome of the experiment ( i.e a typical elements of the sample space ) will be denoted by 𝝎 or е. Thus X(𝝎) represents the real number which the random variable X associates with the outcome 𝝎 .

✦ The value of the random variable will be denoted by small letters x,y,z,... etc. 

SOME PROPERTIES OF RANDOM VARIABLE :

✦ If  X₁ , X₂  are random variables and C is a constant , then CX₁ , X₁ + X₂ , X₁ - X₂ , X₁ * X₂  also random variable.

✦ If X is a random variable , then

• 1/X , where (1/X ) ( 𝝎 ) = ∞ if X(𝝎) = 0.
• X + 𝝎 = max[ 0 , X(𝝎) ]
• X - 𝝎 = min [ 0 , X(𝝎) ]
• | X |

are also random variables.

✦ If  X₁ , X₂  are random variables ,then (i) max[ X₁ , X₂ ] & (ii) min[ X₁ , X₂ ] are also random variables .

✦ If X is a random variable and ∱(.) is a continuous function , then ∱(X) is a random variable.

✦ If X is a random variable and ∱(.) is a increasing function , then ∱(X) is a random variable.

✦ If ∱ is a function of bounded variations on every finite interval [a,b] and X is a random variable, then ∱(X) is a random variable.