The probability distribution function
J. Christopher Clement
School of Electronics Engineering
VIT University
January 28, 2015
VIT
U n i v e r s i t y
(Estd.u/s 3 ofUGC Act 1956)
Vellore– 632 014, Tamil Nadu, India
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An overview of the probability density function (pdf), which is a fundamental concept in probability theory and statistics. The pdf is defined as the derivative of the distribution function, and it has several important properties, such as being non-negative and having an integral of 1 over the entire range of the random variable. The document also includes an example of a triangular pdf and a problem where the student is asked to find the density function given the distribution function. The content covers key topics in probability and statistics, making it potentially useful for university students studying courses related to these fields, such as mathematics, engineering, or data science.
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The probability distribution function
J. Christopher Clement
School of Electronics Engineering
VIT University
January 28, 2015
VIT
U n i v e r s i t y (Estd. u/s 3 of UGC Act 1956)
Vellore – 632 014, Tamil Nadu, India
Outline
Probability Density Function
J. Christopher Clement — The probability distribution function 2/
Probability density function
I (^) The probability density function (pdf) is defined as the derivative of
the distribution function. f X
(x) =
dF X
(x)
dx
. f X
(x) is called as simply
the density function.
Probability density function
I (^) The probability density function (pdf) is defined as the derivative of
the distribution function. f X
(x) =
dF X
(x)
dx
. f X
(x) is called as simply
the density function.
I (^) Properties
- f X
(x) ≥ 0 for all x
∞ ∫
−∞
fX (x)dx = 1
- FX (x) =
x ∫
−∞
fX ()d
- Pr {x 1 < X ≤ x 2 } =
x 2 ∫
x 1
fX (x)dx
Probability density function contd
I (^) A triangular probability density function is given by
fX (x) =
0 ; 3 > x ≥ 13
x − 3
; 3 ≤ x < 8
x − 8
; 8 ≤ x < 13
Find probability that X is
greater than 4.5 but not greater than 6.
I
0 5 10 15
0
x
f
X
(x)
Pr { 4. 5 < X ≤ 6. 7 } =
7 ∫
5
x − 3
dx = 0. 2288
Problem
I (^) A random variable X is known to have a distribution function
F
X
(x) = u(x)
[
1 − e
−x
2
b
]
, where b > 0 is a constant. Find its density
function