MA 2631 Probability Theory Assignment 8

MA 2631
Probability Theory
Assignment 8
due on Tuesday, September 28
based on Lectures of Chapter 4.4–5.2
1. Assume that you have three four-sided dice with number 1, 2, 3, and 4 on the four sides
and let denote by X the sum of the numbers shown on their bottom side. Write down
and sketch the probability mass function and the cumulative distribution function of X.
2. Calculate and sketch the cumulative distribution function (cdf) of a geometric random
variable (i.e., the pmf is given by p(n) = (1 − p)
np for non-negative integers n and some
parameter p ∈ (0, 1)).
3. You arrive at a random time at a bus stop, and you know that the bus is arriving every
30 minutes. Denote with Y the random variable describing your waiting time.
a) What is the probability that you will have to wait longer then 10 minutes.
b) Assume that you waited already 10 minutes, what is the probability that the bus
will arrive in the next 10 minutes?
Describe every of the statements of a) and b) in terms of Y and calculate the
probabilities explicitly using the density fY .
4. Assume that a random variable X has a density of the form fX(x) = cg(x) for some real
constant c and
g(x) =



0 x < 0,
x 0 ≤ x < 5,
10 − x 5 ≤ x < 10,
0 x ≥ 10.
2
a) Determine the value of the constant c, sketch the function fX.
b) What is P[3 ≤ X ≤ 8]?
c) Calculate an sketch FX the cumulative distribution function of X.
5. Assume that a random variable Y has a density of the form fX(x) = cg(x) for some real
constant c and
g(x) =



0 x < 0,
sin(x) 0 ≤ x < π,
0 x ≥ π.
a) Determine the value of the constant c, sketch the function fX.
b) What is P

X ≥
π
6
 
  X ≤
2π
3

?
c) Calculate an sketch FX the cumulative distribution function of X.
6. Let X be a continuous random variable with density
f(x) = 
cx2
e
−x
if x ∈ [0, 1]
0 else.
a) Determine the value of the constant c
b) Calculate the expectation of X.
8 points per problems
Standard Section 2.2: 15, 22, 23, 24; Section 3.1: 1, 2, 5, 8, 11