Finding average using central limit theorem.

Question

I'm have a lot of trouble trying to wrap my head around how exactly to use the central limit theorem in this question.

Image a gambling casino consisting of 100 busy roulette tables. Suppose that each table brings in an average hourly profit of $50 with a standard deviation of 25.

(a) What is the average hourly profit that the entire casino brings in? What is the standard deviation?

(b) Find the probability that the casino will make a profit less than $4500 on any given hour.

I know in this situation I am supposed to CLT to figure out the average profit, but I don't understand how to use the CLT on a continuous case like this.

Answer 1

So we have a hundred tables $X_1, \dots, X_{100}$. Because we have more than lets say $30$ tables, or stochastic processes, we can use the CLT and we know that $\frac{1}{100}(X_1 + \dots + X_{100}) \sim N(50,\frac{25^2}{100})$. All these tables, or $X_i$'s are stochastic with $E(X_i) = 50$ and $V(X_i) = 25^2$.

To calculate the average hourly profit of the entire casino we have to look at $E(X_1 + \dots + X_{100}) = E(X_1) + \dots + E(X_{100}) = 100*50 = 5000$. Because the operator $E$ is linear. For the standard deviation of the whole casino we'd have to look at

$$ V(X_1 + \dots + X_{100}) = V(X_1) + \dots + V(X_{100 }) $$ Since each table is independent of the other tables. So we get:

$$ V(X_1) + \dots + V(X_{100}) = 100*25^2 $$ So the standard deviation is $10*25 = 250$.

So now we know $X_1 + \dots + X_{100} \sim N(5000,250)$.

For part b) you have to calculate $P(X_1 + \dots + X_{100} \leq 4500).$ Now try to use $X_1 + \dots + X_{100} \sim N(5000,250)$ to solve this.

Answer 2

Hint: By the CLT, the distribution of the profit for whole casino follows a normal law of mean $5000\$/$hour and standard deviation $250\$/$hour. $4500$ is $2\sigma$ from the mean.