1-Continuous Random variables and probability density function 

There  exist  random  variables  whose  set  of  possible  values  is uncountable.

Two  examples  are  the  time  that  a  train  arrives  at  a specified stop and the lifetime of a transistor.

Let X be such a random variable. We say that X is a continuous random variable if there exists a nonnegative function f ,

defined for all real 𝑥 ∈ (−∞,∞) having the property that, for any set B of real numbers, 

The function f is called the probability density function of the random variable X. 

In words, the last equation states that the probability that X will be in B may be obtained by integrating the probability density function over the set B.

Since X must assume some value, f must satisfy 

All probability statements about X can be answered in terms of f . For instance, letting B = [a, b], we obtain 

If we let a = b in the last Equation, we get 

In  words,  this  equation  states  that  the  probability  that  a  continuous random  variable  will 

assume  any  fixed  value  is  zero.  Hence,  for  a continuous random variable,

Suppose  that X is  a  continuous  random  variable  whose  probability density function is given by 

(a) What is the value of C?

(b) Find P{X > 1}.

(a)Since f is  a  probability  density  function,

we  must have  implying that 

The amount of time in hours that a computer functions before breaking down is a continuous random variable with probability density function given by 

What is the probability that

(a) a computer will function between 50 and 150 hours before breaking down?

(b) it will function for fewer than 100 hours?

(a) Since

Hence, the probability that a computer will function between 50 and 150 hours before breaking down is given by 

(b) Similarly, 

In other words, approximately 63.3 percent of the time,

a computer will fail before registering 100 hours of use.

If X is continuous with distribution functionand density function ,

find the density function of Y = 2X.