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Chapter 4: Q.4.8 (page 173)

Let B(n,p)represent a binomial random variable with parameters nand p. Argue that

P{B(n,p)鈮�i}=1-P{B(n,1-p)鈮�n-i-1}

Hint: The number of successes less than or equal to iis equivalent to what statement about the number of failures?

Short Answer

Expert verified

Events B(n,p)鈮�i and B(n,1-p)鈮�n-i are equivalent.

Step by step solution

01

Step 1:Given information

Let B(n,p)represent a binomial random variable with parameters nand p. Argue that

P{B(n,p)鈮�i}=1-P{B(n,1-p)鈮�n-i-1}

02

Step 2:Explanation

Assume thatB(n,p)is counter of successes and B(n,1-p)is simply n-B(n,p). Observe that the required equality can be written as

P(B(n,p)鈮�i)=P(B(n,1-p)>n-i-1)=P(B(n,1-p)鈮�n-i)

It is enough to show that events B(n,p)鈮�iand B(n,1-p)鈮�n-iare equivalent.

The event B(n,p)鈮�imeans that we have obtained less or equal to isuccesses. So, that is equivalent to the information that we have obtained more or equal to n-ifailures, which is the second event. Hence, we have proved the claimed.

03

Step 3:Final answer

Events B(n,p)鈮�iandB(n,1-p)鈮�n-i are equivalent.

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