Introduction for discrete random variable example

Introduction for discrete random variable example:

              In the mathematics a random variables is a variable whose value is a function of the outcome of a statistical experiment. Random variables are used in the study of probability. They were developed to get in the analysis of the games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions. Further formalizations have been firmly grounded the entity in the theoretical domains of mathematics by making use of measure theory.

Examples for Discrete Random Variable:

 Discrete Random Variable If a random variable takes only a finite or a countable number of values, it is called a discrete random variable

 The number of heads obtained when two coins are the tossed is a discrete random variable as X assumes the values 0, 1 or 2 which form a countable set. Number of Aces when ten cards are drawn from a well shuffled pack of 52 cards.
The random variable X assumes 0, 1, 2, 3 or 4 which is again a countable set. X (No aces) = 0, X (one ace) = 1, X (two aces) = 2,x (three aces) = 3, X (four aces) =4
Example for Discrete Random Variable Example:

Example:  A random variables X has the following probability mass function

X             1    1     2   3    4     5     6

P(X = x)   k    3k   5k 7k   9k   11k 13k

(1) Find k.

(2) Evaluate P(X < 4), P(X = 5) and P(3< X = 6)

(3) What is the smallest value of x for which P (X = x) >1/ 2 .

Solution:

(1) Since P(X = x) is a probability mass function `sum_(n=0)^6` P(X = x) = 1

ie.,P(X=1) + P(X = 1) +P(X = 2) +P(X = 3) +P(X = 4) +P(X = 5)+P(X = 6) = 1.

? k + 3k + 5k + 7k + 9k + 11k + 13k = 1 ? 49 k = 1 ? k =1 / 49

 (2) P(X < 4) = P(X = 1) + P(X = 1 ) + P(X = 2) + P(X = 3) =1 / 49 +3 / 49 +5 / 49 +7 / 49 =16 / 49

P(X = 5) = P(X = 5) + P(X = 6) =11 / 49 +13 / 49 =24 / 49

P(3 < X = 6) = P(X = 4) + P(X = 5) + P(X = 6) =9 / 49 +11 / 49 +13 / 49 =33 / 49

(3) The minimum value of x may be determined by trial and error method.

P(X = 1) =1 / 49 <1 1="1" 2="2" 49="49" br="br" p="p">
P(X = 2) =9 / 49 <1 2="2" 3="3" 49="49" 6="6" br="br" p="p">
P(X = 4) =25 / 49 > 1 / 2

? The smallest value of x for which P(X = x) > 1 / 2 is 4.