Question

If ( P(x=1) = P(x=2) ) for a distribution of Poisson random variable ( X ), find the mean, variance, and coefficient of skewness of the distribution.

a) Mean: ( 1.5 ), Variance: ( 1.5 ), Coefficient of skewness: ( 0 )
b) Mean: ( 1 ), Variance: ( 1 ), Coefficient of skewness: ( 1 )
c) Mean: ( 2 ), Variance: ( 2 ), Coefficient of skewness: ( 0 )
d) Mean: ( 2 ), Variance: ( 1 ), Coefficient of skewness: ( 0 )

Answer 1

The mean, variance, and coefficient of skewness for a Poisson random variable with equal probabilities at X=1 and X=2 is 1. This is because, for a Poisson distribution, the mean equals the variance, and the coefficient of skewness is 1/√λ, which all result in 1 when λ=1.

Step-by-step explanation:

The student has asked about the mean, variance, and coefficient of skewness for a Poisson random variable (X) where P(X=1) = P(X=2). To find these values, we can solve the Poisson probability function.

For a Poisson distribution, we know that mean (µ) = variance (σ²). Since the probabilities are equal for X = 1 and X = 2, we can set up the following equation using the probability mass function of the Poisson distribution:

e-λλ1 / 1! = e-λλ2 / 2!

Solving this gives us λ = 1. If λ = 1, then both the mean and variance are also 1. The coefficient of skewness for a Poisson distribution is 1/√λ, which would also be 1 in this case.

Hence, the correct answer is (b) Mean: (1), Variance: (1), Coefficient of skewness: (1).