Question 1. Consider the probability density function for the random variable X below,

()={/2 10. 3/8 0<2

1. i) Sketch the density function in a graph.
2. ii) Show that the area under the density function above the horizontal axis over the domain of X ie. [-1, 2] is equal to 1 by making use of the formula for the area of a triangle (base*height/2).
3. iii) What is (>1)? What is (1<0)? What is (=2)?
4. iv) Write down the expression for [] in terms of two separate integrals that treat the intervals [-1,0] and [0,2] separately. (You do not need to make do the integration, just write down the formula with as much of the information provided as possible).
5. v) Explain why []>0 by reference to the concept of the expected value of a continuous random variable and the probability density function graphed in i).
6. vi) Explain why [] is more than the variance of ~(0,2).
7. vii) Find the variances of the uniform random variable in vi).