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This question is from schaums probability and statistics: probability function f(x)= c/3^x, {x=1,2,3...} determine the constant c and find the distribution function. The answer in the back of the book is c=2 but I get 2/3 since:

\sigma$$\sigma$$ 1/3^x = 3/2 I calculate c=2/3 not 2

The answer in the back of the book for the distribution function is F(x) is = 0 for x<1 = 1 - 3^-y y < x < y+1 {y=1,2,3...}

I integrated f(x) by dx and got - 3^-x / ln(3)enter preformatted text here

Nick
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billnyeguy
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1 Answers1

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We Need $1=P(\Omega)=\sum_x\frac{c}{3^x}=c\sum_x\frac{1}{3^x}$. Now for a geometric series starting at x=1, the sum is $\frac{\frac{1}{3}}{1-\frac{1}{3}}=\frac{1}{2}$. Hence, $1=\frac{c}{2}\implies c=2$

As for cdf, you were integrating when you should be summing. This is a discrete distribution, so you want $F(x)=P(X\leq x)=\sum_i P(X=i)=\sum_i\frac{c}{3^i}$, where $i$ ranges from $1$ to $m$. I'm at little confused as to what you wrote, as you said $F(x)=0$ for $x<1$, which is trivial since the minimum value of $x$ is $1$. So if I misunderstood what you were saying in that second part, please let me know and I'll fix my answer

user153582
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  • excellent, why don't you add a word on the cdf part of the question? :-) – Math-fun Oct 30 '15 at 04:09
  • the F(x)=0 for x<1 is a formality, just talking about x>1. For discrete functions, is it always true that F(x) is always equal to $\Sigma$ f(x). The answer schaums gives for F(x) is 1 - 3^y, how does that conform to $\Sigma$ c/3^i. – billnyeguy Oct 30 '15 at 18:31