Continuous random variable

Continuous random variable

Definition 6.5

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

Examples of 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.

Probability density function

Definition 6.6

The probability that a random variable X takes a value in the interval [t₁ , t₂ ] (open or closed) is given by the integral of a function called the probability density function f _X (x) :

P ( t₁ ≤ X ≤t₂ )= ^t2∫_t1  f _X (x)dx .

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 (x) or simply by f (x) must satisfy the following conditions.

(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) = 5x ⁴ ,0 ≤ x ≤ 1

Find a₁ and a₂ such that (i) P[X ≤ a₁ ] = P[X > a₁]         (ii)  P[X > a₂ ] = 0.05

Solution

Continuous distribution function

Definition 6.7

If X is a continuous random variable with the probability density function f_X (x), then the function F_X (x) is defined by

F_X (x) = P[X ≤ x ] = ^x∫_-∞ f (t )dt , −∞ < x < ∞  is  called  the  distribution  function  (d.f)  or sometimes the cumulative distribution function (c.d.f) of the continuous random variable X

Properties of cumulative distribution function

The function F_X(x) or simply F(x) has the following properties

(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