subject: Probability Density Function Random Variable [print this page] Introduction to Probability Density Function Random variable
A random variable X on a sample space S is a function that assigns a real numbers to each sample point in a sample space of Random experiment.
Example: -In a Random experiment of tossing two coins, we have two sample space HH, HT, TH, TT
We consider a random variable X which is the number of heads obtained then X is a random variable can take values 0, 1, 2.
HH HT TH TT
2 1 1 0
Random variables are of two types
1. Discrete Random variable
2. Continuous Random variable
Discrete Function Random Variable of Probability Density
If the Random variable X assumes only finite or countable infinite set of values it is known as discrete random variable.
Probability density function of Discrete Random variable:-
Suppose X is a Random variable which can take at the most a countable number of values X1, X2, X3, Xnwith each value ofX. We associate a number
Pi= P ( X = Xi) ; i = 1,2,..............n
Which is known as the probability of Xiand satisfies the following conditions?
pi= P ( X = Xi)'>='0 ( i = 1,2,..............n ) i.e., pi's are all non- negative and
sum'pi= p1+ p2+................... + pn= 1
i.e., the total probability is one.
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The function pi= P ( X = Xi) ; i = 1,2,..............n is called the probability function or more precisely probability mass function of the random variable X .
Probability Density Function of Continuous Random Variable
A random variable X is said to be continuous if it can take all possible values between two limits.
Probability density function of Continuous Random variable:-
Let X be a continuous random variable taking values on the interval [ a, b ].
A function p ( x ) is said to be probability density function of a continuous random variable X if it satisfies the following properties
p ( x )'>='0 for all x in the interval [a,b]
For two distinct numbers c and d in the interval [a,b]
