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Hello I am wondering about how to approach the following question;

Suppose we are told that the weight of each gum ball ( in centigram) is given by the gamma distribution function, with $\alpha=25$ and $\beta=2$.

We are wanting to know the probability that 100 gum balls will go over the limit of a standard package, of sixty grams.

My thought;

I know that form this type of distribution we have $E[X]= \alpha\beta=50$ Centigrams=$0.5g$

But then how would I go from here. Would I do something like

$P[100X \gt 60]=1-P[x \le 0.6]$ and then actually evaluate the distribution function? Or is there a much simpler way that I am just not saying?

Quality
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2 Answers2

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Indeed, the problem reduces to evaluate a distribution, but you haven't settled it right. Note that you don't have the same gum repeated 100 times. You have 100 of them, each with its own weight. So you have 100 of iid gamma distributed random variables, each with parameters $\alpha=25$ and $\beta=2$.

Brandon
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According to your notation, $E[X]= \alpha\beta$. Then I will assume this means that $\text{Var}[X] = \alpha\beta^2.$ If each weight $X_i$ is $iid$, then let $S = X_1+\dotsb+ X_{100}$. Then $E[X_i] = .5\text{ g}$, $\text{Var}[X_i] = 1 \text{ g}^2$, and $$E[S] = 100E[X_1] = 50$$ and $$\text{Var}[S] = 100\text{Var}[X_1] = 100,$$ using independence. Then the problem becomes the approximation \begin{align*} P(S >60) &= 1-P(S<60)\\ &= 1-P\left[Z<\frac{60-50}{\sqrt{100}}\right] \\ &=1-\Phi(1)\\ &= 0.1586553, \end{align*} where $Z$ is a standard normal distribution.

Em.
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