statistics and probability

Question

After Bank A announced the creation of a new cryptocurrency, EBOCOIN, several analysts began to develop models that aim to explain the evolution of the price of this cryptocurrency over time. Analyst George decided to start his investigation with a very simple model: every day, the price of EBOCOIN has an equal chance of rising by $1 or falling by $1. Knowing that the current price of EBOCOIN is $100, what alternative is the correct one according to George's model?

a) The average price of EBOCOIN in 30 days is $90.

b) The probability of the EBOCOIN price reaching zero for the first time in exactly 50 days is equal to 1/(2⁵⁰).

c) The probability of the EBOCOIN price reaching zero for the first time in exactly 100 days is equal to 1/(2¹⁰⁰).

d) The price of EBOCOIN is equally likely to be $95 or $115 in 20 days.

e) An investor who buys EBOCOIN today can expect positive returns.

Answer 1

First of all, any statements about the price can be made only for the first 100 days, because the description does not clarify what happens once the price becomes 0.

a)

No, the distribution of prices will always have the average of $100.

b)

No, the probability is 0.

The lowest price reachable in 50 days is $50.

c)

Yes, it will happen if every of the 100 days the price looses $1.

d)

Yes, both probabilities are 0.

After an even number of days the price will always be even.

So, it is impossible to be $95 or $115 after 20 days.

e)

He is free to expect anything he likes,

but the expected value of the price will always be $100.