Probability density function can be defined as shown below. Consider a variable 'Y' of a fix range randomly. All possible variables cannot be expressed and hence 'Y' is defined as continuous variable that changes randomly. That is, if ‘f’ is a non negative function which is defined for all 'y', such that:
P (Y ε A) = ∫A f (y) d y,
For any Set A, function f(y) is called probability density function of Random Variable 'Y'. Here 'A' is the set of Real Numbers. Lets discuss some properties of probability density function for random variable 'Y' and constants 'p' and 'q'.
1) P(p ≤ Y ≤ q) = ∫ f (y) d y defined between ‘p’ and ‘q’.
2) ∫ f (y) d y = 1 defined between - ∞ and +∞.
3) P (Y = p) = 0 and,
4) f (y) ≥ 0 for all real 'y'.
We know that negative exponential function is usually represented in following form e (-y). Let’s find out the Probability Density Function Of A Negative Exponential Function.
The p.d.f. (probability density function) of a negative exponential function will be given by following formula.
f (y, ∂) =
Here ∂ > 0 and is usually called as rate parameter. Exponential distribution exists on interval 0 and ∞. Here function f (y, ∂) is called as probability density function. Whenever value of random variable is less than zero, density function becomes zero