1. 2008 Probability Distributions

2. Question One Sillicom find from their broadband customer service hotline that 15% of their broadband customers have problems with the initial setup on their computers. Find the probability that out of the next 10 phone calls to the hotline, less than half will be having problems with the initial setup on their computer

3. Binomial Distribution P(X < 4; n = 10, π = 0.15) = 0.9901

4. Question 2 One of the operators for the hotline finds that on average they are receiving 24 phone calls per hour. The number of phone calls the operator is receiving can be modelled by a Poisson distribution. Calculate the probability that the operator will receive at least 1 phone call in any given 10 minute interval.

5. Poisson Distribution = 24 per hour = 4 per ten minute interval P(X ≥ 1) = 1 – P(X = 0) = 1 – 0.0183 = 0.9817

6. Question 3 Sillicom find that the speed of their wireless routers is normally distributed with a mean of 54 Mbps, megabytes per second, and a standard deviation of 4.1 Mbps. a)Find the probability that a randomly selected router will operate at a speed between 50 Mbps and 60 Mbps.

7. Normal Distribution P( 50 < X < 60) = P( -0.976 < Z < 1.463) = 0.7638 (0.76369 GC)

8. Question 3b a)A router is deemed to be defective if it operates slower than a specific speed. It is found that 2.5% of their routers are defective. Calculate the maximum speed of a defective router.

9. Inverse Normal P(X < k) = 0.025 Z = -1.96 k = 45.96 Mbps

10. Question 4 It is found that the average time spent on hold when calling the hotline is normally distributed with mean 3 minutes and standard deviation 45 seconds. If each call is independent, find the probability that out of the next 4 callers, none of them will be on hold for more than 4 minutes.

11. Normal Distribution P(X > 4) = P(Z > 1.333) = 0.0913 (0.0912 GC) Binomial Distribution P(X = 0; n = 4, π = 0.0913) = 0.6818 (0.6821 GC)

12. Question 5 Sillicom investigated how much time is spent on calls made by their customers on a Sillicom cellphone plan. It was found time spent on calls made to other cellphones in a month is normally distributed with mean 150 minutes and standard deviation 22 minutes and time spent on calls made to landline numbers in a month is normally distributed with mean 80 minutes with standard deviation 15 minutes. You may assume the time spent on landline calls is independent from the time spent on cellphone calls.

13. a)Calculate the probability that a randomly chosen customer on a Sillicom cellphone plan will spend less than 4 hours in total making calls during a month. Sum of two normally distributed independent random variable T = X + Y E(T) = 150 + 80 = 230 min σ(T) = P(X < 240) = P(Z < 0.376) = 0.6465 (0.6464 GC)

14. a)Cellphone customers on a Sillicom plan pay a monthly fee of $55. If they were on pre-pay, they would be charged $0.25 per minute to another cellphone and $0.40 per minute to a landline number. Find the proportion of customers who would be better off on pre-pay than a plan.

15. Linear combination of independent random variables = 69.5 σ(C)= = 8.139 P(X < 55) = P(Z < -1.782) = 0.0373 (0.0374 GC)

16. Question 6 Sillicom have been monitoring their internet connections and notice there are a random number of faults detected every hour. They are concerned because they find that 98% of the time there will be at least one fault detected each hour. Find the probability that in any given hour there will be 5 or fewer faults detected. You must state at least two assumptions you have made.

17. Poisson Distribution Two assumptions from  ｧ faults occur at random  ｧ Each fault is independent  ｧ Faults cannot occur simultaneously  ｧ The probability of a fault is proportional to the length of the time interval.

18. Inverse Poisson P(X = 0) = 1 – 0.98 = 0.02 λ = 3.912 P(X< 5) = 0.7987