A random variable X which can take on any value (integral as well as fraction) in the interval is called continuous random variable.

**Continuous
random variable**

A random variable *X*
which can take on any value (integral as well as fraction) in the interval is
called continuous random variable.

·
The amount of water in a 10 ounce bottle.

·
The speed of a car.

·
Electricity consumption in kilowatt hours.

·
Height of people in a population.

·
Weight of students in a class.

·
The length of time taken by a truck driver to go from Chennai to
Madurai, etc.

The probability that a
random variable *X* takes a value in the interval [*t*_{1} , *t*_{2}
] (open or closed) is given by the integral of a function called the
probability density function
*f _{X} *(

*P *(* t*_{1}* *≤* X *≤*t*_{2}* *)=* ^{t2}*∫

Other names that are
used instead of probability density function include density function,
continuous probability function, integrating density function.

The probability density
functions *f* * _{X}* (

(i)*
f *(* x*)* *≥* *0* *∀* x *and

(ii) ∞∫_{-∞}* f *(*x*)*dx *=* *1* *.

**Example 6.8**

A continuous random
variable X has the following p.d.f

*f *(*x*)* *=* ax *, 0* *≤* x *≤* *1

Determine the constant *a*
and also find *P *[*X*
≤ 1/2]

*Solution:*

**Example 6.9**

A continuous random
variable *X* has p.d.f

*f*(*x*) =
5*x* ^{4} ,0 ≤
*x* ≤ 1

Find *a*_{1}
and *a*_{2} such that (i) *P*[*X*
≤ *a*_{1} ] = *P*[*X* > *a*_{1}] (ii) *P*[*X* > *a*_{2} ] = 0.05

*Solution*

If *X* is a
continuous random variable with the probability density function *f _{X}*
(

*F _{X} *(

The function *F _{X}*(

(i) 0 ≤ *F* (*x*) ≤ 1, − ∞ < *x* < ∞

(ii)

(iii) *F*(⋅)* *is a monotone,
non-decreasing function; that is,* F *(*a*)≤* F *(*b*)* *for*
a *<*b *.

(iv) *F*(⋅)* *is continuous
from the right; that is,

(v)

(vi)

*dF*(*x*)* *is
known as probability differential of* X *.

(vii)

= *P* (*X* ≤ *b*) − *P*(*X* ≤ *a*)

= *F* (*b*) − *F* (*a*)

**Example
6.10**

Suppose, the life in
hours of a radio tube has the following p.d.f

Find the distribution
function.

*Solution:*

**Example
6.11**

The amount of bread (in
hundreds of pounds) x that a certain bakery is able to sell in a day is found
to be a numerical valued random phenomenon, with a probability function
specified by the probability density function *f* (*x*) is given by

(a) Find the value of A.

(b) What is the
probability that the number of pounds of bread that will be sold tomorrow is

(i) More than 10 pounds,

(ii) Less than 10
pounds, and

(iii) Between 5 and 15
pounds?

*Solution:*

(a) We know that

(b) (i) The probability that the
number of pounds of bread that will be sold tomorrow is more than 10 pounds is
given by

(ii) The probability that the number of pounds of bread that will be sold tomorrow is less than 10 pounds, is given by

(iii) The probability that the number of pounds of bread that will be sold tomorrow is between 5 and 15 pounds is

Tags : Definition, Formulas, Solved Example Problems , 12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation

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12th Business Maths and Statistics : Chapter 6 : Random Variable and Mathematical Expectation : Continuous random variable | Definition, Formulas, Solved Example Problems

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