1. Two friends (A and B) agree to meet on 4:00 PM. A usually arrives between 5 minutes early and 5 minutes late. B usually arrives between 5 minutes early and 15 minutes late. Their times of arrival are independent from each other.

   1. What is the probability that B arrives definitely later than A?
   2. What is the expected time that A waits B?
   3. What is the probability that both meet early?

   a) \(P(B > 5) = \dfrac{5 - (-5)}{15 - (-5)} = 1/2\)

   b) \(E[B] - E[A] = 5 - 0 = 5\)

   c) \(P(B < 0, A < 0) = P(B < 0) P(A < 0) = 5/20 * 5/10 = 1/8\)

2. There are three computers, which provide answers to questions with speed according to exponential distribution with means (\(1/\lambda\)) 6, 4 and 3 per hour, respectively. What is the probability that at least one machine provides an answer within the first hour?

\[P(X < x) = 1 - e^{-\lambda x}\]

\[P(X > x) = e^{-\lambda x}\]

The solution is 1 - no machine provides an answer within the hour \(1 - P(X_1 > 1, X_2 > 1, X_3 > 1)\).

\[1 - P(X_1 > 1, X_2 > 1, X_3 > 1) = 1 - e^{-\lambda_1 - \lambda_2 - \lambda_3}\]

    1 - pexp(1,rate=3+4+6)

## [1] 2.260329e-06

3. Time between customer arrivals in a cafe is exponential with the mean value of 6 minutes.

1. What is the probability that no customers arrive in 15 minutes?

\(P(X>15) = 1 - P(X < 15) = 1 - (1 - e^{-\lambda x}) = e^{-15/6} = 0.08\)

2. What is the inter-arrival time if the probability of a customer to arrive is 0.9?

\(P(X > t) = e^{-t/6} = 0.1\), then \(t = 13.81\).

3. What is the probability that 10 customers arrive in the first hour?

\(\lambda * t = 1/6 * 60 = 10\)

\(P(X=10) = \dfrac{e^{-\lambda *t}(\lambda t)^10}{10!} = 0.125\).

4. What is the probability of getting the first customer in 15 minutes if no customer arrived in the first 10 minutes?

\(P(X < 15 | X > 10) = 1 - P(X > 15 | X > 10) = 1 - P(X > 5) = 1 - e^{-5/6} = 0.565\) (memoryless property)

Hint: Check the relationship between Poisson and Exponential distributions.

4. A pack of flour contains 1 kg of flour. Though a flour pouring machine has a standard deviation of 50 gr.

   1. What is the probability that a randomly selected package contains between 925-1075 grams of flour?
   2. If a proper flour package should contain between 1000-x and 1000+x grams of flour, what should x be that 80% of the packages are deemed proper?
   3. Your customer strictly declared that 95% of the packages should contain at least 1000 grams of flour, so you should adjust the mean value. What should be the new mean value?

We have \(\mu = 1000,\ \sigma = 50\). Define \(\Phi(.)\) as the cdf of the standard normal distribution.

1. \(\Phi(\dfrac{1075-1000}{50})-\Phi(\dfrac{925-1000}{50})\)
2. Find an x that \(\Phi(\dfrac{1075-1000}{50})-\Phi(\dfrac{925-1000}{50}) = 0.8\) approximately. The answer is more like how many standard deviations. We can find it by searching for a quantile of \(\alpha=(1-0.8)/2 = 0.1\). Then find the \(y\) \(\Phi(y)=1-\alpha\). \(y = 1.282\) so reverse the procedure from standard normal to N(\(\mu = 1000,\ \sigma = 50\)) by \(y * \sigma\) to find \(x\).
3. Your \(\Phi(\dfrac{1000 - \mu}{50}) = 0.05\). The quantile value for 0.05 is -1.645. So, 1000 - 50*(-1.645) = 1082.25.

#a
pnorm(1075,mean=1000,sd=50)-pnorm(925,mean=1000,sd=50)

## [1] 0.8663856

#b
qnorm(0.9,sd=50)

## [1] 64.07758

#c
1000-qnorm(0.05,sd=50)

## [1] 1082.243

5. There are two different roads to get to Sarıyer. Road A takes 35 minutes on average with standard deviation 5 minutes. Road B takes 32 minutes on average with standard deviation 8 minutes.

   1. Which road has the higher advantage if one wants to reach Sarıyer in 42 minutes?
   2. What is the maximum time of arrival with 90% probability? Calculate for each road.

a) For Road A it will take 42 minutes or less with probability \(\Phi(\dfrac{42-35}{5}) = \Phi(1.4) = 0.92\). For Road B, \(\Phi(\dfrac{42-32}{8}) = \Phi(1.4) = 0.89\). Take Road A. b) It is the quantile value of \(\alpha = 0.9\). Then we need to find \(\Phi^{-1}(\alpha) = 1.28 = \dfrac{x-\mu}{\sigma}\). For Road A \(1.28*5 + 35 = 41.4\), B \(1.28*8 + 32 = 42.2\).

Note: b is phrased weakly. It is also possible to understand it as a probability interval as the probability of an event occurring in normal distribution happens in intervals (i.e. originates from mu). In that case the quantile max value is \(\Phi^{-1}(0.95)=1.64\) as there should be 5% slack at each side of the distribution.