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April 20th, 2018, 01:01 AM   #1
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Cumulative function question

A random variable has a cumulative distribution function

\[F_{X}(x)=c+de^{\frac{-x^2}{2}}\] if x>0 and 0 otherwise, what values must c and d have to make this a valid pdf?
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April 23rd, 2018, 07:08 AM   #2
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The "cumulative probability function", $\displaystyle F_X(x)$, gives the probability that x< X. Now, you are told that the probability that x is negative is 0 so what is the probability that x< 0? What is $\displaystyle F_X(0)$ by this formula? Set those equal. Further, the total cumulative probability is 1. What is the limit of $\displaystyle F_X(x)$ as x goes to infinity? Set that equal to 1. Now you have two equations to solve for p and q.
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April 23rd, 2018, 12:20 PM   #3
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$F_X(\infty)=c=1,\ F_X(0)=c+d=0$
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