Ch 11 - 1, 9

11.1

Question

Pasted image 20241106220442.png

Proof

a. Continuous-time, continuous space.

b. Continuous-time, discrete space

c. Discrete-time, continuous space.

d. Discrete-time, discrete space.

☐

11.9

Question

Pasted image 20241106220601.png

Proof

a. It is a discrete-space rv. Xn is based on number of wins which is an integer.

b. One possible example could be winning on every other spin (starting on a win).

Xn 1 1 2 2 ... 4 5 5
n 1 2 3 4 ... 8 9 10
This is modeled by Xn=⌈12n⌉. Another possiblity could be that they win every 3rd time:
Xn 0 0 1 1 1 2 2 2 3 3
n 1 2 3 4 5 6 7 8 9 10
Modeled by Xn=⌈13(n−1)⌉.

c. See Binomial Experiment and The Binomial Random Variable and Distribution for the idea behind the calculations. In short:

Xn∼Bin(n,1838)

☐