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The waiting time of a customer in a customer service telephone line in company number 1 has the exponential distribution with an expected value of 2.2 minutes.

The waiting time of company 2 has the uniform distribution between 1.5 minutes and 3 minutes.

The waiting time in company 3 has the normal distribution with an expected value of 2.2 minutes and standard deviation of 0.2 minutes.

The customers and the companies are all independent.

A researcher chooses randomly one of 3 couples: Company 1 and company 2 ; Company 1 and company 3 ; Company 2 and company 3.

Within each couple the researcher examines the waiting time of 32 customers for each company in the couple.

It is given that the waiting time of the 32 customers in the company with a lower number was lower than the waiting time of the 32 customers in the company with the higher number (e.g., T1<T2, T1<T3).

What is the probability that the researcher chose the couple (1,2)?

When trying to solve this problem I thought to use the central limit theorem because N>30 for all companies. Therefore, the sum of all waiting times within a company should be roughly normally distributed.

The start should be to do the conditional probability formula:

$P((1,2) | \sum_{Time}^{}C< \sum_{Time}^{}D) $

where C is the company with a lower number and D with a larger number.

If X is the first random variable then:

$X:Exp(\frac{1}{2.2})$ $E(X)=2.2$ $V(X)=4.84$

If Y is the second random variable then:

$Y:U(1.5,3)$ $E(X)=2.25$ $V(X)=0.1875$

If T is the second random variable then:

$T:N(2.2,0.2^{2})$

I am not sure how to continue, how to set the expected value and standard deviation of the normal distribution under the central limit theorem and to calculate the probability.

Thank you !!

BlueSigma
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    The application of the CLT to the (skewed) Exponential distribution is doubtful. Analyze it instead by noting that sums of iid Exponential distributions are Gamma distributions. For details on the sum of iid Uniform distributions, see https://stats.stackexchange.com/questions/41467. – whuber May 27 '22 at 13:44
  • In fairness, gamma with a shape parameter of 32 is pretty damn close to normal. Similarly, Irwin-Hall with parameter of 32 is also damn close to normal. I know if I were doing this problem I'd lean on the CLT to simplify it. – Ben May 27 '22 at 14:54
  • If $X_1, X_2$ have $E(X_i) = \mu,$ then $E(X_1+X_2) = 2\mu.$ If $V(X_1)=V(X_2) = \sigma^2$ and $X_1,X_2$ indep., then $V(X_1+X_2) = 2\sigma^2.$ // Use that idea to find mean and variance for sums of $32.$ Then use CLT to assert approx normality (exact normality if $X_i$ normal), // By simulation, aprx means and SDs are 70.4, 12.46 for 1; 72.0, 2.45 for 2; 70.4, 1.13 for 3. Can you roughly match those using above formulas? – BruceET May 27 '22 at 17:40

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