Let X Geom(p). Let s 0 be an integer. a. Show that P(X > s) = (1 p)s . (Hint: The

Chapter 4, Problem 25

(choose chapter or problem)

Let \(X \sim \operatorname{Geom}(p)\). Let \(s \geq 0\) 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 \(t \geq 0\) be an integer. Show that \(P(X>s+t \mid X>s)=P(X>t)\). This is called the lack of memory property. [Hint: \(P(X>s+t \text { and } X>s)=P(X>s+t)\).]

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:

X \sim Geom(p)  

S \geq 0  

P(X>s)=(1 - p)s  

t \geq 0  

P(X>s+t | X>s)=P(X>t)

P(X>s + t and X>s)=P(X>s+t)