Exercise Solution 3.4

In the following solution, ϕ_Z denotes the PDF of Z.

1. By [3.6],

   [s1]

   

   [s2]

   

   [s3]

   

   [s4]

   

   [s5]

   

   [s6]

   
2. By [3.6] and [3.7],

   [s7]

   

   [s8]

   

   [s9]

   

   [s10]

   

   [s11]

   

   [s12]

   

   [s13]

   

   [s14]

   
3. The standard deviation of a random variable is simply the square root of its variance:

   [s15]