math probability question help

UCLA Mathematics 174E Fall 2017

Assignment #1 D ue M o nday , Octo ber 9 , 2 0 1 7 , in lecture

Reading • Review your notes and references from your prerequisite probability theory class.

o For probability theory reference material, consult the notes of Bertsekas and Tsitsiklis,

available here.

o My colleague, Professor Heilman, has written notes on the above material, available

here.

• The notes of Professor Žitković on stochastic processes are available here.

o Chapter 4 has a nice introduction to random walks

Problems Carefully solve the following problems.

1. Let 𝑥 be a standard normally distributed random variable with probability density

𝑝(𝑥) = 1

√2𝜋 𝑒

− 𝑥2

2

Show that

a. 𝐸(𝑥) = 0

b. 𝐸(𝑥2) = 1

c. 𝐸(𝑒 𝜎𝑥 ) = 𝑒 𝜎2

2

d. Find a formula for 𝔼(𝑥𝑛 ) where 𝑋 is standard normal. Prove your formula is correct.

2. For a random walk with Gaussian steps find 𝑃(𝑥1 ≥ 0, 𝑥2 ≥ 0)

3. Consider a random walk 𝑥𝑛 with independent, identically distributed increments 𝑑𝑛.

a. Calculate 𝐶𝑜𝑣(𝑥𝑛 , 𝑥𝑚 ) for 𝑚 ≤ 𝑛 in the case that the steps are standard normally

distributed.

b. For 𝑘 < 𝑙 < 𝑚 < 𝑛 show that the increments 𝑥𝑙 − 𝑥𝑘 and 𝑥𝑛 − 𝑥𝑚 are independent. (Don’t

assume that the 𝑑𝑖 have any specific distribution.)

c. Let 𝑘 < 𝑙 and suppose that the value of 𝑥𝑘 is known. Show that 𝑥𝑙 is then independent of 𝑑𝑖

for 𝑖 < 𝑘. (Similarly, don’t assume that the 𝑑𝑖 have any specific distribution.)

4. Consider the exponential random walk model for two stocks 𝑆 and 𝑆 ′. Assume that both have

Gaussian increments and that they have the same growth rate and volatility; i.e.

𝜇 = 𝜇′ and 𝜎 = 𝜎′

Also, assume that the size of the time steps 𝑑𝑡 and 𝑑𝑡′ are related by the equation,

𝑑𝑡 = 2𝑑𝑡′

Let 𝑛′ = 2𝑛 and show that 𝑆𝑛 and 𝑆𝑛′′ have the same probability density function.

5. Let 𝑋, 𝑌 be independent, discrete random variables. Show that

𝑃(𝑋 + 𝑌 = 𝑧) = ∑ 𝑃(𝑋 = 𝑥)𝑃(𝑌 = 𝑧 − 𝑥)

𝑥∈ℝ

, ∀𝑧 ∈ ℝ

6. Let 𝑋, 𝑌 be independent, continuous random variables with densities 𝑝𝑋 , 𝑝𝑌 respectively. Let

𝑝𝑋+𝑌 denote the density function of the sum 𝑋 + 𝑌. Show that

𝑝𝑋+𝑌(𝑧) = ∫ 𝑝𝑋 (𝑥)𝑝𝑌(𝑧 − 𝑥)𝑑𝑥 , ℝ

∀𝑧 ∈ ℝ

7. Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble

either wins $1 or loses $1 independent of the past with probabilities 𝑝 and 𝑞 = 1 − 𝑝

respectively. Let 𝑅𝑛 denote the total fortune after the 𝑛 th gamble. The gambler’s objective is to

reach a total fortune of $𝑁, without first getting ruined (running out of money). If the gambler

succeeds, then the gambler is said to win the game. In any case, the gambler stops playing after

winning or getting ruined, whichever happens first.

a. What is the probability that the gambler has not lost after 5 gambles?

b. Suppose that 𝑝 = 0.5 and 𝑁 = 5. What is the probability that the gambler has won by

the 11th gamble?

8. Estimate the probability that 1,000,000 coin flips of fair coins will result in more than 501,000

heads, using the Central Limit Theorem.