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If I have a random variable $Y=a(\frac{b}{c}+\frac{X}{c})$, where $a,b,c$ are all integers and $X$ is Poisson distributed with a mean which is large enough to be approximated by the normal distribution. Can $Y$ then be approximated by a normal distribution as well?

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    $\begingroup$ Yes. Why would you think otherwise? $\endgroup$ – Hong Ooi Feb 6 '15 at 13:37
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Your formulation is equivalent to $Y=a_1X+b_1$; you don't really need three coefficients to achieve translation and scaling. This is more general than your formulation (because my $a_1$ and $b_1$ are not restricted to integer multiples of $1/c$ for $c$ integer) but the answer applies more generally.

Can Y then be approximated by a normal distribution as well?

If you look at the cdf of the original Poisson and its normal approximation (i.e. draw a plot of the two cdfs), then this becomes obvious:

CDFs of: 1. Poisson with its normal approximation; 2. Rescaled Poisson with its normal approximation

All translation and scaling does is change the labels on the x-axis!

Algebraically it's clear why this must be so by considering the probability statements involving the two discrete cdfs (showing the height of the cdf at corresponding points is the same) and similarly for the corresponding normal approximations.

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