Let A be the smallest field over the л-system P. Use the inclusion-exclusion formula (2.2) to show that probability measures agreeing on P must agree also on A.
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- Suppose the probability of erroneously transmitting a single digit is P=0.03. Compute the probability of transmitting a 4-bit code word with (a) at most one error, and (b) exactly four errors.1. Let X ~ Poisson(A) and Y ~ Poisson(u). Assume that X and Y are independent. Use probability generating functions to find the distribu- tion of X + Y.1. Show that the collection of random variables on a given probability space and having finite variance forms a vector space over the reals.
- Mead. ASSume that the amounts of Weight the male College Student gain during their freshmen year are normallu distiibuted with a medn of N= 12K4 and a standard deviaton f O= 5:3 k4 complete parts (0)throug (C) If Imale college Student is randomly selected, find Vhe probability Hrat he gaines between okg and 3 kg during Freshmen year. The probability isSuppose that on a particular power grid, 30 wind power generating stations each have output which is uniformly distributed between 24 and 36 MW. a) What is the (approximate) probability that the total power available from these 30 stations would exceed 950 MW? Assume that the power generated by each station is independent of other stations (which, of course, doesn’t make much sense, but taking into account dependence needs to know about the covariance between stations, so we’ll ignor that for now). b) What total power level from these 30 stations would have only a 5% chance of being exceeded, approximately?Suppose we randomly draw two integers from the range [1, n] with uniform probability. Define X to be the value of the first integer drawn; define Y to be the value of the second integer drawn. Define Z = |X - Y|. Compute E(Z).
- Suppose that the probability is 0.63 that a car stolen in a certain Western city will be recovered. Use the com-puter printout of Figure 1 to find the probability that at least 8 of 10 cars stolen in this city will be recov-ered, using (a) the values in the P(X = K) column;(b) the values in the P(X LESS OR = K) column.Assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 34.Complete parts (a) through (c) below. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.Suppose X is a discrete random variable with finite first and second moments. Show that E (x -)E(x-A) for all b E R.
- i) Let X be an integrable random variable on (2, F, P) and G be a sub o-field of F. State the definition of the conditional expectation of X given G.Assume that X is a hypergeometric random variable with N=50, S= 20, n=5. Calculate the following probabilities. A) P(X=2) B) P(Xgreaterthanorequalto2) C) P(Xlessthanorequalto3)An environmental engineer collected 10 moss and 10 lichen specimens. The engineer instructs a laboratory intern to randomly select 15 of the specimens. The probability mass function of the number of lichen specimens selected at random is: a) H (x; 15; 10; 20) b) P (x; 10) c) Bnegative (x, 10, 0.666) d) B (x, 10, 0.666)