GATE EC 2016 Set 1 — Question 12
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Engineering Mathematics → Statistics & Distributions → Poisson Distribution
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Question
The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is
Correct answer
0.9 to 1.1
Solution
Let be a Poisson-distributed random variable with parameter (mean) . The probability mass function of is given by:For a Poisson distribution, the mean and variance are both equal to :The second moment of , denoted as , is related to the mean and variance by the formula:Substituting the known values into this relation:We are given that the second moment . Substituting this into the equation yields a quadratic equation in terms of :Factoring the quadratic equation:This gives two possible roots:Since the parameter of a Poisson distribution must be strictly positive (), we discard the negative root. Thus:The mean of the random variable is .Correct Answer: 1 (which lies within the allowed range of 0.9 to 1.1).
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