In this section we learn (i) Continuous random variable (ii) Probability density function (iii) Distribution function (Cumulative distribution function). (iv) To determine distribution function from probability density function. (v) To determine probability density function from distribution function.

**Continuous
Distributions**

In this
section we learn

(i) Continuous
random variable

(ii) Probability
density function

(iii) Distribution
function (Cumulative distribution function).

(iv) To
determine distribution function from probability density function.

(v) To
determine probability density function from distribution function.

Sometimes
a measurement such as current in a copper wire or length of lifetime of an
electric bulb, can assume any value in an interval of real numbers. Then any
precision in the measurement is possible. The random variable that represents
this measurement is said to be a **continuous** random variable. The range of the
random variable includes all values in an interval of real numbers; that is,
the range can be thought of as a continuum of real numbers

** **

*Let **S** be a sample space and
let a random variable X *:* S *→* **ℝ** that takes on any value in a set **I **of **ℝ** . Then **X** is called a **continuous random variable** if P *(*X = x*) *= *0* for every x in **I*

** **

*A non-negative real
valued function f *(* x*)* is said to be a **probability density
function** **if, for each possible
outcome x, x *

A function *f* (.) is a probability
density function for some continuous random variable *X* if and only if it satisfies the following properties.

(i) *f *(*x*)*
*≥* *0* *, for every* x* and (ii) ^{∞}∫_{-∞}*f *(*x*)*dx* = 1 .

It
follows from the above definition, if *X*
is a continuous random variable,

*P*(*a ≤ X ≤ b*) = ^{b}∫* _{a} f*(

That is
probability when *X*
takes on any one particular value is zero.

** **

*The **distribution function** ***or*** cumulative distribution function F *(

**Remark**

(1) In
the discrete case, *f (a) = * *P *(*X = a*) is the probability that *X* takes the value *a*.

In the
continuous case, *f *(*x*) at *x* = *a* is not the
probability that *X* takes the value *a*, that is *f *(*a*) ≠ *P *(*X
= a*) . If *X* is continuous type, *P *(*X
= a*) = 0 for *a* ∈ ℝ.

(2) When
the random variable is continuous, the summation used in discrete is replaced
by integration.

(3) For
continuous random variable

P(*a* < *X* < *b*) = P(*a* ≤ *X*
< *b*) = P(*a* < *X* ≤ *b*) = P(*a* ≤ *X* ≤ *b*)

(4) The
distribution function of a continuous random variable is known as Continuous
Distribution Function.

** **

For a
continuous random variable X, the cumulative distribution function satisfies
the following properties.

(i) 0 ≤
*F* ( *x*) ≤ 1 .

(ii) *F *(*
x*)* *is a real valued
non-decreasing. That is, if* x *<* y *, then* F *(* x*)* *≤* F *(*
y*)* *.

(iii) *F *(*
x*)* *is continuous everywhere.

(iv) lim
_{x → −∞} *F* (*x*) = *F*(
− ∞) = 0 and lim_{ x → ∞} *F* (*x*) = *F*
(+∞) = 1.

(v) *P *(*X
>* *x*) = 1 − *P *(*X≤
x*) = 1 − *F *(*x*)* *.

(vi) *P*(*a <* *X < b*)* = F*(*b*)*
−F*(*a*)* *.

** **

**Example 11.11**

Find the constant *C*
such that the function is a density function, and compute (i) *P*(1.5 < *X*
<
3.5) (ii) *P* ( *X* ≤ 2) (iii) *P*(3 <
*X* ) .

**Solution**

Since the given function is a probability density function,

From the given information,

Since *f* ( *x*) is continuous, the probability that *X* is equal to any particular value is
zero.

Therefore when the random variable is continuous, either or
both of the signs < by ≤ and > by ≥
can be interchanged. Thus

** **

Both the probability density function and the cumulative
distribution function (or distribution function) of a continuous random
variable *X* contain all the
probabilistic information of *X* . The *probability distribution *of* X *is determined by either of them. Let
us learn the method to determine* *the
distribution function *F* of a
continuous random variable *X* from the
probability density function *f *(*x*)*
*of* X *and vice versa.

** **

**Example 11.12**

If* X *is the random
variable with probability density function *f*(*x*) given by,

**Check:**

(i) Whether *F* ( *x*) is continuous everywhere.

(ii) From the Fig. 11.16, triangle area =
1/2 *bh* = 1.

** **

Let us learn the method to determine the probability density
function *f* ( *x*) from the distribution function *F* ( *x*) of a continuous
random variable *X* *.*

*Suppose **F** *(* **x*)* is the distribution function of a continuous random variable **X** . Then the probability
density function **f*(*x)** is given by*

* f*( *x*) = *dF*(*x*)
**/ ***dx = F* ′ ( *x*) *, wherever derivative exists.*

** **

If *X*
is the random variable with distribution function F (*x*) given by,

then find (i) the probability density function *f* (*x*)
(ii) P ( 0.2 ≤ X 0.7).

**Solution**

** **

Let *X*
be a random variable denoting the life time of an electrical equipment having
probability density function

Find (i) the value of *k* (ii) Distribution function (iii) P (*X*< 2)

(iv) calculate the probability that *X* is at least for four unit of time (v)
P (*X* = 3) .

Tags : Definition, Properties | Probability Distributions | Mathematics , 12th Maths : UNIT 11 : Probability Distributions

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12th Maths : UNIT 11 : Probability Distributions : Continuous Distributions | Definition, Properties | Probability Distributions | Mathematics

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