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December 4th, 2012, 07:00 PM   #1
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probability at a point for a continuous random variable

Hello all,
I am new to this forum, and also to the field of random variables. I have a doubt regarding the existence of the probability that a continuous random variable x to take a particular value X. I read that, this probability is equal to 0, which was proved by considering the probability,
P(X-e<x<=X) = F(X) - F(X-e), where F(x) is cumulative distribution function, and e is a small positive number. Since F(x) is continuous, when the limits are taken, to tend e->0,
the LHS becomes P(x=X) , and RHS, goes to zero, as the left limit Lt(e->0) F(X-e) = F(X).

But i cant interpret this result. Wont the probability density function p(x) at the point x=X, give the probability that x takes the value X? Why would this be equal to 0?. Plz help me out...
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December 5th, 2012, 12:24 PM   #2
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Re: probability at a point for a continuous random variable

Probability density function is the derivative of the probability distribution function. Unless the distribution has a jump at a particular point, the probability of that point = 0.
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January 7th, 2013, 02:27 AM   #3
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Re: probability at a point for a continuous random variable

Hello mathman, thanks for the reply...
Is the definition of density function is : the probability that the random variable takes the value? In that case, i dont understand why the probability a random variable takes a particular value is zero? Or is my definition of density function wrong ? Kindly help me out...
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January 7th, 2013, 03:25 PM   #4
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Re: probability at a point for a continuous random variable

Quote:
Originally Posted by rsashwinkumar
Hello mathman, thanks for the reply...
Is the definition of density function is : the probability that the random variable takes the value? In that case, i dont understand why the probability a random variable takes a particular value is zero? Or is my definition of density function wrong ? Kindly help me out...
The best way to look at probability is in terms of a distribution function, say F(x). (For simplicity, I will take the case where there are F(x) is continuous.) The probability that the random variable is in an interval (a,b) is F(b) - F(a). Here the probability at a point c = F(c) - F(c) = 0. The probability density is the derivative of F(x) and should not be interpreted a a probability.

The only case where probability is non-zero at a point is where F(x) has a jump at that point and the probability is the value of the jump.
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