Question

Does a continuous random variable X exist with E(X − a) = 0, where a is the
day of your birthdate? If yes, give an example for its probability density function.
If no, give an explanation.
(b) Does a continuous random variable Y exist with E
(Y − b)
2

= 0, where b is
the month of your birthdate? If yes, give an example for its probability density
function. If no, give an explanation.

Answer 1

• 
  a is a day of a birthday. Or, in other words, any number belonging to R+.
  This can exist.
  
  An example to site this is:
  
  
  It is known that, in such a case,
  
  
  
  
  
• This event cannot happen.
  This is because,
  
  a is a month of a birthday. Or, in other words, any number belonging to R+.
  
  
  Now,
  Also,
  
  
  Then, integral over such a non-negetive random variable can never be 0, unless,
  Again, this will happen, only if,
  
  
  However, for a continuos random variable, there does not exist any positive probability to any fixed point.
  
  Hence, the condition cannot be true for any continuos random variable.
  
  

I hope this clarifies your doubt. If you're satisfied with the solution, hit the Like button. For further clarification, comment below. Thank You. :)