sorry for my ignorance!

is it possible to calculate a cumulative probability for an event to occur knowing a mean, that a certain percent of values will fall below a threshold, and assuming a normal distribution?

for example:

say we had a mean of 35000, and knew that 10% of values were below 1000. can we calculate the probability that x is less than or equal to 1350?

  • $\begingroup$ Not answering your question, but just a comment: If your variable X is always positive (thus 0% of values below 0) then your assumption of a normal distributed variable would be very wrong. $\endgroup$ – Martijn Weterings Mar 5 '18 at 11:04

Say the RV $X$ is $\sim N(35000, \sigma^2)$. Then $\frac{X - 35000}{\sigma} \sim N(0, 1)$.

Note that $$X < 1000 \Leftrightarrow \frac{x - 35000}{\sigma} < {-34000 \over \sigma},$$

so 0.1 is the probability of a standard normal variable to be less than ${-34000 \over \sigma}$.

Using a table for the cumulative function for the standard normal distribution, a probability of 0.1 is obtained approximately at -1.2816. Therefore,

$$ \sigma \simeq {34000 \over 1.2816} . $$

Your variable is therefore approximately $X \sim N(35000, ({34000 \over 1.2816})^2)$. Any numerical software will tell you the probability of its being over 1350.


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