1. Show
that Y = n if and only if I1 = 0, ..., In
- 1 = 0, In = 1.Virtual Laboratories > Bernoulli Trials > 1 2 3 [4] 5 6 7
Suppose again that our random experiment is to perform Bernoulli trials I1, I2, ... with parameter p in (0, 1]. In this section we will study the random variable Y that gives the trial number of the first success. Recall that Xn, the number of successes in the first n trials, has the binomial distribution with parameters n and p.
1. Show
that Y = n if and only if I1 = 0, ..., In
- 1 = 0, In = 1.
2. Use the
result of Exercise 1 and independence to show that
P(Y = n) = p(1 - p)n - 1 for n = 1, 2, ...
The distribution defined by the density in Exercise 2 is known as the geometric distribution with parameter p.
3. In the
negative binomial experiment, set k = 1. Vary p with the
scroll bar and note the shape of the density function. With p = 0.2, run the
simulation with an update frequency of 10. Watch the apparent convergence of the relative
frequency function to the density function.
4. Show directly
that the geometric density function really is a density function.
5. A fair die is
tossed until an ace occurs. Find the probability that the die will have to be tossed at
least 5 times.
The following exercises give the mean, variance, and probability generating function of the geometric distribution.
6. Show that E(Y)
= 1 / p.
7. var(Y)
= (1 - p) / p2.
8. Show that E(tY)
= pt / [1 - (1 - p)t] for |t| < 1 / (1 - p).
9. In the
negative binomial experiment, set k = 1. Vary p with the
scroll bar and note the location and size of the mean/standard deviation bar. With p
= 0.4, run the simulation with an update frequency of 10. Watch the apparent convergence
of the sample mean and standard deviation to the distribution mean and standard
deviation..
10. A type of
missile has failure probability 0.02. Give the mean and standard deviation of the number
of launches before the first failure.
11. Show that the
conditional distribution of Y given Xn = 1 is uniform on {1,
2, ..., n}. Note that the distribution does not depend on p. Interpret the
result probabilistically.
12. A student
takes a multiple choice test with 10 questions, each with 4 choices. The student blindly
guesses and gets one question correct. Find the probability that the correct question was
one of the first 4.
The following problems explore a very important characterization of the geometric distribution.
13. Suppose that Z
is a random variable taking positive integer values. Show that Z has the
geometric distribution with parameter p if and only if
P(Z > n) = (1 - p)n for n = 0, 1, 2, ...
14. If Z
has a geometric distribution, show that Z satisfies the memoryless property:
for positive integers n and m,
P(Z > n + m | Z > m) = P(Z > n)
15. Conversely,
show that if Z is a positive integer-valued random variable that satisfies the
memoryless property, then Z has a geometric distribution.
16. Show that Z
has the memoryless property if and only if the conditional distribution of Z - m
given Z > m is the same as the distribution of Z.
17.
In the negative binomial experiment, set k = 1, p = 0.3. Run the experiment 1000
times, with an update frequency of 100. Compute the appropriate relative frequencies and
empirically investigate the memoryless property
P(Y > 5 | Y > 2) = P(Y > 3)
The memoryless property has important implications in gambling.
18. Recall
that an American roulette wheel has 38 slots: 18 are red, 18 are black, and 2 are green.
Suppose that you observe red on 10 consecutive spins. Give the conditional distribution of
the number of additional spins needed for black to occur.
We will now explore another gambling situation, known as the Petersburg problem, which leads to some famous and surprising results. Suppose that we are betting on a sequence of Bernoulli trials with success parameter p > 0. We can bet any amount of money on a trial at even stakes: if the trial results in success, we receive that amount, and if the trial results in failure, we must pay that amount. We will use the following strategy, known as a martingale strategy:
19. Let V
denote our net winnings when we stop. Show that V = c.
Thus, V is not random and V is independent of p > 0! Since c is an arbitrary constant, it would appear that we have an ideal strategy. However, let us study the amount of money W needed to play the strategy.
20. Show that W
= c(2Y - 1).
21. Use the result in the previous exercise to show that
if p 
1 / 2.Thus, the strategy is fatally flawed when the trials are unfavorable and even when they are fair.
22. Compute E(W)
explicitly if c = 100 and p = 0.55.
23. In the
negative binomial experiment, set k = 1. For each of the following values of p,
run the experiment 100 times, updating after each run. For each run compute W (with
c = 1). Find the average value of W over the 100 runs:
For more information about gambling strategies, see the chapter on Red and Black.
A coin has probability of heads p in (0, 1]. There are n players who take turns tossing the coin in round-robin style: player 1 first, then player 2, ..., then player n, then player 1 again, and so forth. The first player to toss heads wins the game.
Let Y denote the number of the first toss that results in heads. Of course, Y has the geometric distribution with parameter p. Additionally, let W denote the winner of the game; W takes values 1, 2, ..., n. We will compute the probability density function of W in two different ways
24.
Show that for i = 1, 2, ..., n,
W = i if and only if Y = i + kn for some k = 0, 1, 2, ...
That is, using modular arithmetic, W = (Y - 1) (mod n) + 1.
25. Use the result of the previous exercise and the geometric distribution to
show that
P(W = i) = p(1 - p)i - 1 / [1 - (1 - p)n] for i = 1, 2, ..., n
26.
Argue that P(W = i) = (1 - p)i
- 1P(W = 1). Use this result to re-derive the
probability density function in the previous exercise.
27.
Explicitly compute the probability density function of W when the
coin is fair (p = 1/2) in each of the following cases
Note from Exercise 25 that W itself has a truncated geometric distribution.
28. Show that the distribution of W is the same as the conditional
distribution of Y given Y
n:
P(W = i) = P(Y = i
| Y
n ) for i = 1, 2, ..., n.
29.
Show that for fixed p in (0, 1], the distribution of W
converges to the geometric distribution with parameter p as n
.
30. Show that for fixed n, the distribution of W converges
to the uniform distribution on {1, 2, ..., n} as p
0.
31. What happens in the game when p = 0? Compare with the limit in
the previous exercise.