Let X ~ Geom(p). Let s ? 0 be an integer.a. Show that P(X

Chapter 4, Problem 25SE

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QUESTION:

Let X ∼ Geom(p). Let s ≥ 0 be an integer.

a. Show that \(P(X>s)=(1-p) s\). (Hint: The probability that more than s trials are needed to obtain the first success is equal to the probability that the first s trials are all failures.)

b. Let t ≥ 0 be an integer. Show that \(P(X>s+t \mid X>s)=P(x>t)\). This is called the lack of memory property. [Hint: P (X > s + t and X > s) = P (X > s + t).]

c. A penny and a nickel are both fair coins. The penny is tossed three times and comes up tails each time. Now both coins will be tossed twice each, so that the penny will be tossed a total of five times and the nickel will be tossed twice. Use the lack of memory property to compute the conditional probability that all five tosses of the penny will be tails, given that the first three tosses were tails. Then compute the probability that both tosses of the nickel will be tails. Are both probabilities the same?

Equation transcription:

Text transcription:

P(X>s)=(1-p) s

P(X>s+t \mid X>s)=P(x>t)

Questions & Answers

QUESTION:

Let X ∼ Geom(p). Let s ≥ 0 be an integer.

a. Show that \(P(X>s)=(1-p) s\). (Hint: The probability that more than s trials are needed to obtain the first success is equal to the probability that the first s trials are all failures.)

b. Let t ≥ 0 be an integer. Show that \(P(X>s+t \mid X>s)=P(x>t)\). This is called the lack of memory property. [Hint: P (X > s + t and X > s) = P (X > s + t).]

c. A penny and a nickel are both fair coins. The penny is tossed three times and comes up tails each time. Now both coins will be tossed twice each, so that the penny will be tossed a total of five times and the nickel will be tossed twice. Use the lack of memory property to compute the conditional probability that all five tosses of the penny will be tails, given that the first three tosses were tails. Then compute the probability that both tosses of the nickel will be tails. Are both probabilities the same?

Equation transcription:

Text transcription:

P(X>s)=(1-p) s

P(X>s+t \mid X>s)=P(x>t)

ANSWER:

Answer:

Step 1 of 3:

(a)

In this question, we are asked to calculate .

Where random variable  follows the geometric distribution .

Here parameter  is probability of success in trials.

Random variable  represent the number of trials up to and including the first success.

Given hint:

The probability that more than s trials are needed to get the first success which is equal to the probability that the first s trials are all failures.

Trials are independent bernoulli trials with the probability of success  and the  probability of failure is .

Therefore by multiplication rule

 = ) = )

                 =  )

    =   ………………….(1)

Hence proved.


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