29650 Engineering Mathematics 2 - Tutorial sheet 1
Question 1
Let P be the probability density function over the set {1, 2, 3, 4, 5, 6} defined by:
P (1) = P (6) = 0.3 (1)
P (2) = P (3) = P (4) = P (5) = 0.1 (2)
Let r be a discrete random variable governed by P. Calculate the expected value E(r) and variance V (r) of r.
Question 2
Let r be a continuous random variable governed by a uniform probability density function over the interval [2, 10]. Calculate the expected value and variance of r.
Question 3
Let r be a continuous or discrete random variable. Show that
V (r) = E(r2 ) - [E(r)]2 (3)
Question 4
Let r1 and r2 be uncorrelated random variables. Show that
V(r1 + r2 ) = V (r1 ) + V (r2 ) (4)
Question 5
Let x be a continuous random variable. Let a, b ∈ R, a > 0. Define a new random variable q by
q = (x × a) + b (5)
Show that E(q) = E(x) × a + b and V (q) = a2 V (x)
Question 6
Let p be the continuous PDF defined by
p(x) = 0, x < 0 (6)
p(x) = 0 ≤ x ≤ π (7)
p(x) = 0, x > π (8)
Let r be a continuous random variable governed by p.
1. Verify that p is a PDF
2. Calculate E(r) and V (r).