Resistors of a certain type have resistances that are exponentially distributed with parameter λ = 0.04. An operator connects 50 independent resistors in series, which causes total resistance in thecircuit to be the sum of individual resistances. Find the joint probability that each and everyone of them has resistance of less than 4.

Show work for every step. Two types of customers (Preferred and Regular) call a service center. Preferred customers call with rate 3 per hour, and Regular customers call with rate 5 per hour. The interarrival times of the calls for each customer type are exponentially distributed. If the call center opens at 8:00 AM, what is the probability that the first call (regardless of customers type who calls) is received before 8:15 AM?

Please answer a,b,c, and show steps if possible. Do not randomly write wrong answers if don't know how to solve.

You are planning to go fishing with two friends. The time (in hours) before you catch a fish is exponentially distributed with parameter 6. How long do you need to arrange for the fishing so that there is a 95% probability that everyone catches at least one fish? [Assume that the events that each person catches a fish are independent.]

The life X (in hours) of a battery in constant use is a random variable with exponential density. What is the probability that the battery will last more than 12 hours (h) if the average life is 8 h?