GATE CS 2017 Set 2 — Question 31
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Engineering Mathematics → Probability & Statistics → Discrete Distributions
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Question
For any discrete random variable , with probability mass function
, and , define the polynomial function
. For a certain discrete random variable , there exists a scalar such that . The expectation of is
, and , define the polynomial function
. For a certain discrete random variable , there exists a scalar such that . The expectation of is
Correct answer
(B) Nβ
Solution
The function is the probability generating function (PGF) of the discrete random variable .
The expectation of a random variable can be found from its PGF using the formula , where is the first derivative of the PGF with respect to .In this problem, we are given the PGF of a random variable as:
To find the expectation of , we first need to compute the derivative of with respect to :
Using the chain rule, we get:
Now, we evaluate this derivative at to find the expectation :
This is also recognizable as the PGF of a binomial distribution , for which the expectation is known to be .Therefore, the expectation of is .
The expectation of a random variable can be found from its PGF using the formula , where is the first derivative of the PGF with respect to .In this problem, we are given the PGF of a random variable as:
To find the expectation of , we first need to compute the derivative of with respect to :
Using the chain rule, we get:
Now, we evaluate this derivative at to find the expectation :
This is also recognizable as the PGF of a binomial distribution , for which the expectation is known to be .Therefore, the expectation of is .
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