Shashank performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of Z fair coins. Given t hat Sh
Shashank performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of Z fair coins. Given t hat Sh
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![the accident.
Question 16
Shashank performs an experiment comprising a series of independent trials. On
each trial, he simultaneously flips a set of Z fair coins.
a. Given that Shashank has just had a trial with Z tails, what is the probability
that next two trials will also have this result?
b. Whenever all the Z coins land on the same side in any given trial, Shashank
calls the trial a success.
i.
Find the PMF for K, the number of trials up to, but not including the
second success.
Find the expectation and variance of M, the number of tails that
occur before the first success where Z = 3 in this case.
ii.
(Hint: Write M = X, + X2 + .…+ Xy Where X; is number of tails in
that occur on unsuccessful trail i and N is number of unsuccessful
trails and find E (M) in terms of E (X) and E(N) and for var(M) first
write in terms of M | N (i.e. M given N) and later write var(M) in
terms of E(N), E (X), var(N), var(X) using the above derived
variance equation in terms of M | N.)
You are expected to write all the steps involved in deriving E (M),
var(M) as mentioned in the hint.
...
c. Sandeep conducts an experiment like Shashank's, except that he uses M
coins for the first trial, and then he obeys the following rule: Whenever all
the coins land on the same side in a trial, Sandeep permanently removes
one coin from the experiment and continues with the trials. He follows this
rule until the (M- 1)h time he removes a coin, at which point the
experiment ceases. Find E[X], where X is the number of trials in Sandeep's
experiment.
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Transcribed Image Text:the accident.
Question 16
Shashank performs an experiment comprising a series of independent trials. On
each trial, he simultaneously flips a set of Z fair coins.
a. Given that Shashank has just had a trial with Z tails, what is the probability
that next two trials will also have this result?
b. Whenever all the Z coins land on the same side in any given trial, Shashank
calls the trial a success.
i.
Find the PMF for K, the number of trials up to, but not including the
second success.
Find the expectation and variance of M, the number of tails that
occur before the first success where Z = 3 in this case.
ii.
(Hint: Write M = X, + X2 + .…+ Xy Where X; is number of tails in
that occur on unsuccessful trail i and N is number of unsuccessful
trails and find E (M) in terms of E (X) and E(N) and for var(M) first
write in terms of M | N (i.e. M given N) and later write var(M) in
terms of E(N), E (X), var(N), var(X) using the above derived
variance equation in terms of M | N.)
You are expected to write all the steps involved in deriving E (M),
var(M) as mentioned in the hint.
...
c. Sandeep conducts an experiment like Shashank's, except that he uses M
coins for the first trial, and then he obeys the following rule: Whenever all
the coins land on the same side in a trial, Sandeep permanently removes
one coin from the experiment and continues with the trials. He follows this
rule until the (M- 1)h time he removes a coin, at which point the
experiment ceases. Find E[X], where X is the number of trials in Sandeep's
experiment.
hp
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