Problem 3 {17 marks]. Let U1 N UNA] a] (7 points) Find and sketch fully labeled graphs of the probability density function and the cumulative density function of U1. b) (4 points) A 4 metre long cord is randomly cut once. Find the probability that one length of cord will be at least twice as long as the shorter length of cord. Hint: Use a diagram to help you map out the pmbabttittes, and make use of uniform distribution pmperttes! c) (6 points) Derive the mean and variance in general for any uniform random variable U2 W UP}, [I]. Problem 4 [25 marks]. Suppose the life expectancy of house cats can be modeled with a normal distribution with an average of 14.5 years and a standard deviation of 3 years. a) {5 points] Find the central range of life expectancies of 95% of house cats. Interpret what this interval means in this context (try not to restate the question). b) (4 points) To what age do the top 2% of cats live? What percentile does this age represent? Explain what your results mean. c) (3 points] What is the probability that a randomly selected house cat will live more than 18 years? d) (4 points) What is the probability that a randomly selected house cat will live more than 18 years if we know it is already 15 years old? Justify in plain English why the result is different from (c). e) (3 points) Consider more generally a random variable X where X m N[a,02). Find the value of k in terms of number of standard deviations 0' such that Flip. also 1': X 5 p. + 390') = 0.95. Comment on any similarities you notice to part (a). f) (2 points) Referring to part [(1], what does the random variable Z, or your Zscores in the normal probability table1 represent in terms of p: and or for a normal random variable X m N(p'., 0'2)? g) {4 points) Using X N N[a,a2), show that P[|X pl