please answer all the questions

Notes: 1. Show all work, points are awarded based on the work provided. 2. When providing probabilities provide them as numbers between 0 and 1 (not as percentages), and keep at least 3 decimal places. 1. You just rented a large house, and the realtor gave you 4 keys, one for each of the 4 doors @pis) of the house. Unfortunately, all the keys look identical, so to open the front door, you must try them at random. a. Assume that after an unsuccessful trial, you mark the corresponding key so that you never try it again. 1 Find the Probability Distribution for the number of trials you need to open the door. This may be a table that contains the possible values and probability associated with each. Hint: check that your probabilities sum to 1. i. Calculate the expected number of trials necessary to open the front door. b. Assume that after an unsuccessful trial, you do not mark the corresponding key thus at each trial you are equally likely to choose any key. 8 Find the Probability Distribution for the number of trials you need to open the door. This may be a rule (a function) since there are now infinite possibilities. What family of distributions does this random variable belong to? 1i. Determine the expected number of trials necessary to open the front door. 2, The number flaws in a fiber optic cable follows a Poisson distribution with an average of @pts) 1.5 flaws per 50m of cable. a. Calculate the probability of exactly three flaws in 150m of cable. b. Calculate the probability of at least two flaws in 100m of cable. c. Calculate the probability of exactly one flaw in the first 50m of cable and exactly one flaw in the second 50m of cable. Hint: the number of flaws in disjoint intervals are independent. 3. The pdf of a random variable, X, is given below: (4pts) fx(x) = kvx for 0 S x $ 4 0 otherwise a. Find the value of k so that fx (x) is a pdf. b. Compute the CDF of X. C. Compute the P(1