In a Party in the campus everyone that comes Gets a CupCake , Only $5$ Students Came.
The Diameter of the cupcake is Exponential Random variable with $E(D) = 15 [cm]$.
What is the Probability that the Average Diameter of the Cupcakes that the students Got is Less Than $14 [cm]$.
Final Answer$: \fbox{0.3192}$
$\fbox{I think i need to use normal dist}$ $\fbox{i know that because when i use the Z table with Z = $-0.47$ I get the correct answer.}$
$1 - P(Z<0.47) = 1- 0.6808 = 0.3192$
I dont know how to approach this question . First i thought using the central limit therom but i got wrong answer i knew that the number of students $n=5$ and it isn't $30\leq n$ thats why its wrong.
The only Thing i did was :
$D_{avg} = \frac{\sum_1^5 D_i}{5}$
$E(D_{avg}) = 15 [cm]$
$Var(D_{avg}) = 45 [cm^2]$
D~$\exp(\frac{1}{15})$
and i dont know how $D_{avg}$ distribute.
The sum of $5$ independent exponentially distributed random variables with mean $\ 15\ $ follows an Erlang distribution with shape parameter $5$ and rate parameter $\ \lambda=\frac{1}{15}\ $: $$ P\left(\sum_{i=1}^5D_i\le x\right)=1-e^{-\lambda x}\sum_{k=0}^4\frac {(\lambda x)^k}{k!}\ , $$ from which it follows that \begin{align} P\left(\frac{\sum_{i=1}^5D_i}{5}\le 14\right)&=1-e^{-\frac{14}{3}}\sum_{k=0}^4\frac {\left(\frac{14} {3}\right)^k}{k!}\\ &\approx 0.4992\ . \end{align} So, unless I'm misunderstanding the problem, either there's something missing from its statement, or the given answer of $\ 0.3192\ $ is incorrect.