Issue: Suppose that a couple will have 3 young people. Acknowledge that the probability of having a young woman is .487 and the probability of having a child is .513. Furthermore expect that x is a self-assertive variable for the amount of young women and find the probability where x is 0, 1, 2, or 3. Every together word, answer the going with probability questions.

a. What is the probability that none of the three children will be young women?

b. What is the probability that correctly one of the three will be youngsters?

c. What is the probability that definitely two of the three will be young women?

d. What is the probability that all of the three of the youths will be young women?

Acknowledge the speed of vehicles along an open stretch of a particular expressway in Texas that isn't energetically journeyed has an around Normal dissemination with a mean of 71 mph and a standard deviation of 3.125 mph.

a) The current posted speed limit is 65 mph. What is the degree of vehicles going over the current posted speed limit?

b) What degree of the vehicles would be going under 50 mph?

c) What degree of the vehicles would be heading off to some place in the scope of 60 and 75 mph?

d) State experts are aware of the road not being seriously managed, and that it can manage a higher speed limit. Regardless, they presently will execute a high cost for speeding over the new speed breaking point to promise some level of as a rule security. Speeds will be checked by radar. Acknowledge a comparable Normal scattering of vehicle speeds continues into the future as already. What should be the new speed limit so much that pretty much 10% of vehicles will speed over the new posted speed limit? Show the sum of your intuition/work in reacting to this.

answer for d

i9i

a) Market research has shown that 60% of individuals who are familiar with a particular thing truly buy the thing. A sporadic illustration of 15 individuals were familiar with the thing.

I. Describe the variable of interest for the present circumstance.

ii. What probability scattering do you think best depicts the condition? Why?

iii. Discover the probability that absolutely 9 will buy the thing.

iv. If 80 individuals are familiar with the thing, choose the amount of person who are depended upon to buy the thing.

b) It is understood that an ordinary of 5 trains go through Grand Central Terminal as expected. Find the probability that

I. Correctly 4 trains will pass rapidly

ii. under 2 trains will pass in an hour

(Ehrenfest's chain) Suppose we have two boxes with total N balls inside. At each step of the Markov chain, one of the balls is chosen at random uniformly and moved from its current box to the other one. Let X\" be the number of balls in the first box at time n. Then the transition matrix defining the Markov chain is Nn, ify=w+1; mN PC1319): W ify=x1; 0, otherwise. Show that the unique stationary distribution arr for this chain is a binomial distribution with parameters N and 1/2, that is, N 1 7r(x)=($)2N, form=0,1,...,N. There is a one-dimensional lattice with lattice constant a as shown in Figure 1. An atom is initially at the origin a = 0 and transits from a site to a nearest-neighbor site every + seconds. The probabilities of transiting to the right and left are p and q = 1-p, respectively. Let n be the number of transitions to the right, and n' the number of transitions to the left. The total number of transitions is thus Nun + n'. O Figure 1: A ID lattice with an atom hopping on it. (a) Where is the atom at f = Nr in terms of N, n, and a? That is, what is the displacement a of the atom after / transitions? (b) What is the probability of the displacement in (a)? (c) What are the possible values of n? (d) What is the average value of n and the displacement a after N transitions? (e) What is the average displacement in one transitions? (f) Compare the results from (d) and (e) and show that they are consistent with the central limit theorem by noting that the total displacement after / transitions is the sum of displacements from each transition, the latter of which is a binomial random variable. (g) What is the dispersion of n and the displacement a of N transitions? (h) What is the dispersion of the displacement in one transition? (i) Compare the results from (g) and (h) and show that they are consistent with the central limit theorem. (i) Apply the central limit theorem to determine the probability distribution of the total displacement a of N transitions when N is large.ror tnis result. 7. A machine consists of two parts that fail and are repaired in- dependently. A working part fails during any given day with probability a. A part that is not working is repaired by the next day with probability b. Let X\" be the number of working parts in day to. (a) Show that Xn is a threestate Markov chain and give its onestep transition probability matrix P. (b) Show that the steady state pmf at is binomial with param eter p = b/(a + b). (c) What do you expect is the steady state pmf for a machine that consists of m parts? 12. a. Binomial Hedging. Consider a one-year European call option with a strike price K= $100 on a stock that has a current price of S, =$100per share. The call option is the right to buy, but not the obligation to buy, at the strike price K. In one year the price of a share will either rise to S, =$120 or fall to S = $80. In general, the up payoff is C, = max(S, - K, 0) and the down payoff is C. = max(S/ -K, 0). Solve the following equations for NVand B: NS, + B = C. NS, + B = Ca where / is the number of shares of stock and B is the maturity value of a zero-coupon bond. Compute the value of the hedging portfolio (which gives the value, or price, of the call option) NS +e"B where the risk-free interest rate is = .05 and? = 1 year in this problem. b. Binomial Pricing. The one-year forward value of the stock is assumed to be Se" = pS, + (1-p)s Calculate the value of p, the risk-neutral up transition probability. This option's value also can be computed as the value Co C=e"[PC, + (1-p)(,] after the risk-neutral up probability p has been computed in Se" = pS, + (1-p)S, . C, is the present value of the expected payoffs of the option. Compute C and thereby confirm that it is equal to the value of the hedging portfolio in part (a). A note on notation: The notation x means x* x if x20 10 otherwise . so, (S, - K)"means the same thing as max(S, - K, 0) and (S, -K )" means the same thing as max(S, - K, 0)