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Chapter 4: Q. 4.27 (page 172)

If Xis a geometric random variable, show analytically that

P{X=n+k∣X>n}=P{X=k}

Using the interpretation of a geometric random variable, give a verbal argument as to why the preceding equation is true.

Short Answer

Expert verified

We proved that

P{X=n+k∣X>n}=P{X=k}

Step by step solution

01

Definition

A random variable that takes the value k, a non-negative integer with probability pk(1-p).

02

Calculation

P{X=n+k∣X>n}=P{X=n+k∩X>n}P(X>n)

P{X=n+k∩X>n}=(1-p)n+k-1p

P(X>n)=First nare failures

=(1-p)n

⇒P{X=n+k∣X>n}=(1-p)n+k-1(1-p)n.p

=(1-p)k-1p

=P{X=k}

Hence proved

03

continue Calculation

Also because trials are independent

Given that first ntrials does'nt result in success

Next kthwill result in success and next (k-1)in failures =(1-p)k-1p

=P{X=k}

⇒P{X=n+k∣X>n}=P{X=k}

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