Probability;

5. Let the continuous random variables X, 1" have joint distribution 1 if\" 1 fart-1111'}: { Girl d: y I: I 1. (a) Determine the constants a and b. (b) Find the pdf of X. Be sure to give a formula for fx(x) that is valid for all r. (c) Calculate the expected value of X. (d) Calculate the standard deviation of X.4. Suppose X is uniform on the interval from 1 to 2. Compute the pdf and expected value of the random variable Y = 1/X.2. Suppose the number of children born in Ruritania each day is a binomial random variable with mean 1000 and variance 100. Assume that the number of children born on any particular day is independent of the numbers of children born on all other days. What is the probability that on at least one day this year, fewer than 975 children will be born in Ruritania?3. Suppose that the time until the next telemarketer calls my home is distributed as an exponential random variable. If the chance of my getting such a call during the next hour is .5, what is the chance that I'll get such a call during the next two hours?7. I repeatedly roll a fair die. If it comes up 6, I instantly win (and stop playing); if it comes up k, for any & between 1 and 5, I wait & minutes and then roll again. What is the expected elapsed time from when I start rolling until I win? (Note: If I win on my first roll, the elapsed time is zero.)5. I toss 3 fair coins, and then re-toss all the ones that come up tails. Let X denote the number of coins that come up heads on the first toss, and let Y denote the number of re-tossed coins that come up heads on the second toss. (Hence 0