solve the following question

A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let Y =the number of forms required of the next applicant. The probability that y forms are required is known to be proportional to y-that is, p(y) = ky for y=1,....,5. a. What is the value of k? [Hint: . y=P(V) = 1. ] b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required? d. Could fp(y) = yz /50 for y = 1......,5 be the pmf of Y?Refer to Exercise 12 and calculate V(Y) and sY. Then determine the probability that Y is within 1 standard deviation of its mean value. Reference exercise -12 Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. a. What is the probability that the flight will accommodate all ticketed passengers who show up? b. What is the probability that not all ticketed passengers who show up can be accommodated? c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?Use the definition in Expression (3.13) to prove that V(aX + b) = 02. ox [ Hint: With h(X) = aX + b, Ejh(X)] = ap = b where p = E(X).] Reference Expression (3.13 The variance of h(X) is the expected value of the squared difference between h(X) and its expected value: Vax + b) = a . of [Hint: With h(X) = aX + b, Ejh(X)] = ap + b where a = E(X).] When h(X) = aX + b, a linear function, VIh(X)] = ohn = Eth(x) - Eth(X]] . p(x) Substituting this into (3.13) gives a simple relationship between V[h(X)] and V(X):According to the article "Characterizing the Severity and Risk of Drought in the Poudre River, Colorado" (J. of Water Res. Planning and Mgmnt., 2005: 383-393), the drought length Y is the number of consecutive time intervals in which the water supply remains below a critical value yo (a deficit), preceded by and followed by periods in which the supply exceeds this critical value (a surplus). The cited paper proposes a geometric distribution with p = .409 for this random variable. a. What is the probability that a drought lasts exactly 3 intervals? At most 3 intervals? b. What is the probability that the length of a drought exceeds its mean value by at least one standard deviation?The Centers for Disease Control and Prevention reported in 2012 that 1 in 88 American children had been diagnosed with an autism spectrum disorder (ASD). a. If a random sample of 200 American children is selected, what are the expected value and standard deviation of the number who have been diagnosed with ASD? b. Referring back to (a), calculate the approximate probability that at least 2 children in the sample have been diagnosed with ASD? c. If the sample size is 352, what is the approximate probability that fewer than 5 of the selected children have been diagnosed with ASD?Consider a collection A1, ..., Ak of mutually exclusive and exhaustive events, and a random variable X whose distribution depends on which of the Ai's occurs (e.g., a commuter might select one of three possible routes from home to work, with X representing the commute time). Let E(x|A; ) denote the expected value of X given that the event A, occurs. Then it can be shown that E(X) = [E(XIA, ).P(A;) the weighted average of the individual "conditional expectations" where the weights are the probabilities of the partitioning events. a. The expected duration of a voice call to a particular telephone number is 3 minutes, whereas the expected duration of a data call to that same number is 1 minute. If 75% of all calls are voice calls, what is the expected duration of the next call? b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type i cookie has a Poisson distribution with parameter p; = i + 1 (i = 1, 2, 3). If 20% of all customers purchasing a chocolate chip cookie select the first type, 50% choose the second type, and the remaining 30%opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?Use a statistical software package to construct a normal probability plot of the tensile ultimate- strength data given in Exercise 13 of Chapter 1, and comment. Reference exercise 13 The accompanying specific gravity values for various wood types used in construction appeared in the article "Bolted Connection Design Values Based on European Yield Model" (J. of Structural Engr., 1993: 2169-2186): 31 .35 .36 .36 37 .38 40 40 40 .41 41 42 42 42 42 42 43 44 45 46 46 47 48 48 48 51 54 54 55 58 62 66 .66 67 .68 .75 132.7 132.9 133.0 133.1 133.1 133.1 133.1 133.2 133.2 133.2 133.3 133.3 1335 133.5 1335 133.8 133.9 134.0 134.0 134.0 134.0 134.1 134.2 134.3 134.4 134.4 134.6 134.7 134.7 134.7 134.8 134.8 134.8 134.9 134.9 135.2 135.2 135.2 135.3 135.3 135.4 135.5 135.5 135.6 135.6 135.7 135.8 135.8 135.8 135.8 135.8 135.9 135.9 135.9 135.9 136.0 136.0 136.1 136.2 136.2 136.3 136.4 136.4 136.6 136.8 136.9 136.9 137.0 137.1 137.2 137.6 137.6 137.8 137.8 137.8 137.9 137.9 138.2 138.2 138.3 138.3 138.4 138.4 138.4 138.5 138.5 138.6 138.7 138.7 139.0 139.1 139.5 139.6 139.8 139.8 140.0 140.0 140.7 140.7 140.9 140.9 141.2 141.4 141.5 141.6 142.9 1434 143.5 143.6 143.8 143.8 143.9 144.1 1445 1445 147.7 147.7 a. Construct a stem-and-leaf display of the data by first deleting (truncating) the tenths digit and then repeating each stem value five times (once for leaves 1 and 2, a second time for leaves 3 and 4, etc.). Why is it relatively easy to identify a representative strength value? b. Construct a histogram using equal-width classes with the first class having a lower limit of 122 and an upper limit of 124. Then comment on any interesting features of the histogram