# BU7527 Example sheet — 2 Mike Peardon — School of Mathematics, TCD

```BU7527
Example sheet — 2
Mike Peardon — mjp@maths.tcd.ie
School of Mathematics, TCD
Tuesday, 29th September
Try to answer these questions before tomorrow’s lectures. We will go through solutions
in class.
1. A fair coin is tossed until it lands heads. The random number N is deﬁned as the
total number of coin ﬂips made. Find
a. the expected value and
b. the variance of N .
You might ﬁnd the following identities useful:
∞
X
k=0
kρk =
ρ
(1 − ρ)2
and
∞
X
k=0
k 2 ρk =
ρ(1 + ρ)
,
(1 − ρ)3
when 0 ≤ ρ &lt; 1
2. In a game, you pay €1 to play and then roll three fair dice. You win €N , where N = 3
if all three dice match, N = 2 if two dice match and N = 0 if all three dice show
diﬀerent numbers. Find the expected value and variance of your winnings.
3.
a. Find fX , the probability density function of the random number X with cumulative distribution function given by



0
x≤0
πx
FX (x) = sin 2 0 &lt; x &lt; 1


1
x≥1
b. Find FY , the cumulative distribution function of the random number Y with
probability density function
fY (y) =









0
y≤0
4y
0 &lt; y ≤ 12
4(1 − y) 12 &lt; y ≤ 1
0
y≥1
1
4. Two random numbers X and Y are both in the range [0, 1] and have a joint pdf given
by
7
fX,Y (x, y) = − (x − y)2
6
Find the probability density function of X alone.
5. Two random numbers X and Y have a probability distribution given by
(
fX,Y (x, y) =
k
0
if x2 + y 2 ≤ 1
otherwise
where k is an unknown constant. Are X and Y independent? Find k and then ﬁnd
the probability P (X 2 + Y 2 &lt; 14 ).
2
```