Math 105/206 - Quiz 5, Mar 27 2015
IMPORTANT: Write your name AND student number somewhere on this sheet.
No calculators, books or notes. Please show your work to get full marks. (10 marks total + 4 bonus marks)
Problem 1
The percentage of fourth-year students that graduates at UBC each year is a random variable X modelled by
a probability distribution function f (x) = 3(1 − x)2 for 0 ≤ x ≤ 1, and 0 otherwise.
• compute the CDF of X. (2 marks)
• compute the expected value of X. (2 marks)
• compute the variance of X. (2 marks)
Problem 2
Decide if each of the following sequences converges or diverges and motivate your decision, and for those that
converge compute the limit. (1 mark each)
n
1
an = −
3
an =
n(3n2 + 2)
n2 + 1
3
an = (2) n
an = en − n2
Problem 3
Show that the decimal number 0.999999
P · · · 9 ·k· · = a0.9 is equal to 1 by writing it as a sum of a geometric series
and using the formula for the sum, ∞
k=0 a · r = 1−r where |r| < 1. (4 marks)
A solution that uses the “multiplication by 10” trick that I showed you in class will be marked with a 0.