Find if a random variable is given by its density function, such that

1. Find , ,  if a random variable  is given by its density  function, such that

, if 

, if 

, if 

1. Let  be given by its distribution function  F(x), such that

F(x) = 0 if x ≤ 0

F(x) = if 0<x≤2

F(x) =1 if x>2

a) Graph the distribution function 

b) Graph the density function

c) Find ,, 

2. A random variable  is distributed normally with E(X) = 8 and σ(X) =3.FindP(9≤X<11).

3. The distribution of the width of a standard piece of computer paper is normal with an expectation of 8.5 inchesand the standard deviation of 0.2 inch.

a) Find the probability that thewidth of any given piece of computer paper is between 8.40and 8.55.

b) Find the probability that the width of any given piece of computer paper is less than 8.35.

c) Find the probability that the width of any given piece of computer paper is greater than 8.6.

4. The density function of a random variable  is given by 

Find its (a) math expectation, (b) variance and (c) distribution function.

5. Find the density function of a normally distributed random variable, if E(X) = 7.8 and σ(X) = 4.1

6. It’s known that a random variable  is distributed normally with   E(X) = 3 and it’s also known  thatp(0≤X≤1)+p(5≤X≤6) = 0.6.  Find p(5≤X≤6).

7. Describe a real-life situation to which the normal distribution could be applied. Explain why.  How does a normal distribution differ from one, which is not? Which aspects of your situation might be altered?