Question

A die is rolled and, independently, a coin is tossed. Let X be the value of the die if the coin is H and minus the value of the die if the coin is T.

(a) Calculate and plot the PMF of X.

(b) Calculate E [X] and var(X)

(c) Calculate and plot the PMF of X 2 − 2X.

2. A drunk walks down a street. Assume he starts at block 0. Every 10 minutes, he moves north a block (+1) with probability p or south a block (-1) with probability 1 − p, independently of any movement outside of those 10 minutes.

(a) What’s the PMF of his position after 10n minutes?

(b) Where would you expect him to be in an hour?

3. Legend has it, a Jinn was fond of probability. Those unfortunate enough to encounter it in the desert, were given an intriguing offer:

In this desert old, Without rain, as told,

Rains one, is arid ten, I will take one of ten,

Give me all your gold, And when rain, behold,

And I will make one ten! What is left is therein.

Distrustful of poetic spirits, travelers were quick to turn down this offer and to continue their journey, amusing the Jinn to no end.

You’ve taken ECE 341 and are not so dismissive. You interpret the fact that “for every 1 day that it rains 10 do not” by the appropriate probability p of rain happening independently every day. You know that if you give x amount of gold to the Jinn, you’ll end up with 10x gold at the beginning. But every day that it doesn’t rain, you will lose 10% of what remains. Until it rains, at which point you get to keep whatever remains. Assume gold can be divided arbitrarily. Starting with x gold, calculate the expected amount of gold left at the end. Do you give your gold to the Jinn?

Answer 1

1. a). Please find the calculated PMF of X

All the possible X's value will have equal probability, with P= 1/12 = 0.083

Die	Coin	X	P (X)
1	H	1	0.083
1	T	-1	0.083
2	H	2	0.083
2	T	-2	0.083
3	H	3	0.083
3	T	-3	0.083
4	H	4	0.083
4	T	-4	0.083
5	H	5	0.083
5	T	-5	0.083
6	H	6	0.083
6	T	-6	0.083

Plot is drawn below :

Below is the table calculated as :

Die	Coin	X	P (X)	X * P(X)	X^2	X^2* P(X)
1	H	1	0.083	0.083	1	0.083
1	T	-1	0.083	-0.083	1	0.083
2	H	2	0.083	0.167	4	0.333
2	T	-2	0.083	-0.167	4	0.333
3	H	3	0.083	0.250	9	0.750
3	T	-3	0.083	-0.250	9	0.750
4	H	4	0.083	0.333	16	1.333
4	T	-4	0.083	-0.333	16	1.333
5	H	5	0.083	0.417	25	2.083
5	T	-5	0.083	-0.417	25	2.083
6	H	6	0.083	0.500	36	3.000
6	T	-6	0.083	-0.500	36	3.000
				0.000		15.167

b). E(X) is given by = Summation (X * P(X)) = 0 ( refer above table in green mark)

Var (X) is given by = E (X^2) - (E(X))^2

E (X^2) = Summation (X^2 * P(X)) = 15.167

Therefore Var (X) = 15.167

c) PMF plot of X^2 - 2X is below :

Die	Coin	X	P (X)	X * P(X)	X^2	X^2* P(X)	2*X	(X^2 - 2*X)
1	H	1	0.083	0.083	1	0.083	2	-1
1	T	-1	0.083	-0.083	1	0.083	-2	3
2	H	2	0.083	0.167	4	0.333	4	0
2	T	-2	0.083	-0.167	4	0.333	-4	8
3	H	3	0.083	0.250	9	0.750	6	3
3	T	-3	0.083	-0.250	9	0.750	-6	15
4	H	4	0.083	0.333	16	1.333	8	8
4	T	-4	0.083	-0.333	16	1.333	-8	24
5	H	5	0.083	0.417	25	2.083	10	15
5	T	-5	0.083	-0.417	25	2.083	-10	35
6	H	6	0.083	0.500	36	3.000	12	24
6	T	-6	0.083	-0.500	36	3.000	-12	48

Let us assume that Y be another random variable defined by Y = (X^2 - 2X)

Below is the calculated table for Y.

Y	P(Y)
0	0.083
-1	0.083
3	0.167
8	0.167
15	0.167
24	0.167
35	0.083
48	0.083

PMF plot for Y is shown below :