I need help for the following questions. Thanks

Question 1

$100 1. Consider a gamble X such that X = with probability 0.6 $100 with probability 0.4 (a) Is this a fair gamble? Verify your answer (b) Suppose that Anh has an initial wealth of $100 and her utility function over money is u (w) = vw. Will she join this gamble? Justify your answer. (c) Suppose that Chris has an initial wealth of $0 and his utility function over money is u (w) = e10. Will he join this gamble? Justify your answer. (d) Suppose that Beth has an initial wealth of $10000 and her utility function over money is u (w) = w. Will she join this gamble? Justify your answer and explain difference from part (b)When supplying labour in the education sector, professors take into account the wage a worker such as themselves could potentially carn in an alternate sector. Let L and If denote the quantities of labour supplied and demanded in education, and let w be the wage in education and Wa be the expected wage in an alternate sector. A. When Wa increases, Is will (increase/decrease/remain unchanged) and the labour supply curve will shift (give direction or write "none" if it will not shift). At the same time. Ld will (increase/decrease/remain unchanged) and the labour demand curve will shift (give the direction or write "none" if it will not shift). B. If Ls = 4w+12-Wa, graph the labour supply curve for Wa=2 and Wa=1/. C. Suppose Ed=100-5was in the previous question. Use part B to show what happens to the education equilibrium wage when alternative wages increase. The new equilibrium has (higher/lower/same) wage and (greater/lower/same) number of professors working in this sector. Explain why.10 W S D O 1 2 3 4 5 6 7 8 9 10 L Click to see additional instructions Suppose there are 10 workers. If this labour market is perfectly competitive, workers are voluntarily unemployed in equilibrium. The number of involuntarily unemployed workers in equilibrium is Workers receive| * of total surplus in equilibrium. Name a wage the union could bargain for which is higher than the perfectly competitive outcome but would reduce worker surplus: $ Name a wage the union could bargain for which is higher than the competitive wage and for which the magnitude of the adverse employment effect is 3: $ Suppose the union is able to raise wages $1 above the competitive level. Worker surplus will now be of (the new) total surplus. The new total surplus is of the original total surplus. Alice? Bob, and lEarl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between I pin and 1:30 pm on that day. 1What is the probability that Carl will have to wait more than 10 minutes for one oi the others to show up? (So consider Carl's waiting time until at least one of the others has arrived.) 6. A stick of length L (a positive constant) is broken at a uniformly random point X. Given that X = r, another breakpoint Y is chosen uniformly on the interval [0, a]. (a) Find the joint PDF of X and Y . Be sure to specify the support. (b) We already know that the marginal distribution of X is Unif(0, L). Check that marginal- izing out Y from the joint PDF agrees that this is the marginal distribution of X. (c) We already know that the conditional distribution of Y given X = x is Unif(0, r). Check that using the definition of conditional PDFs (in terms of joint and marginal PDFs) agrees that this is the conditional distribution of Y given X = x. (d) Find the marginal PDF of Y. (e) Find the conditional PDF of X given Y = y