Assume the random variable X has distribution X ∼ Bin(9, 0.5) and let Y = (−1)^X. Derive the probability mass function of Y . I know how to derive the mean and variance, but how to derive the pmf? Can anyone helps me out?
How to derive pmf from a ramdom variable X to a random variable Y = (-1)^X?
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$\begingroup$
calculus
statistics
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2**Hint:** Try to condition on values $X$ can take. – 2017-01-29
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1i.e. Look up the law of total probability – 2017-01-29
1 Answers
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If $X \sim Binom(9, .5),$ then by symmetry, $P(X \text{ even}) = 0.5.$ [Note that ${a+b \choose a} = {a+b \choose b},$ for non-negative integers $a$ and $b.$]
So $P((-1)^X = 1) = 0.5.$ You can finish it from there.
# Using R statistical software:
sum(dbinom(c(0,2,4,6,8), 9, .5))
## 0.5
