Please do number 6.46 and show work for finding the moment generating function of6.406.416.426.43*6.446.456.466.47function of U = Y² + Y²/².Suppose that Y₁ and Y2 are independent, standard normal random variables. Find the densityLet Y₁, Y2, ..., Y, be independent, normal random variables, each with mean and variance o².U = Σi=1 a¡Y₁.Let a₁, a2, ..., an denote known constants. Find the density function of the linear combinationA type of elevator has a maximum weight capacity Y₁, which is normally distributed with mean5000 pounds and standard deviation 300 pounds. For a certain building equipped with this typeof elevator, the elevator's load, Y2, is a normally distributed random variable with mean 4000pounds and standard deviation 400 pounds. For any given time that the elevator is in use, findthe probability that it will be overloaded, assuming that Y₁ and Y₂ are independent.mean and variance o2.Refer to Exercise 6.41. Let Y₁, Y₂, ..., Y,, be independent, normal random variables, each withaFind the density function of Y =Exercises 323-=nnΣΥi=1Y₁.16 and n =b If o²25, what is the probability that the sample mean, Y, takes on a valuethat is within one unit of the population mean, u? That is, find P(Y- | ≤ 1).c If o² = 16, find P(Y - μ| ≤ 1) if n = 36, n = 64, and n = 81. Interpret the results ofyour calculations.The weight (in pounds) of "medium-size" watermelons is normally distributed with mean 15and variance 4. A packing container for several melons has a nominal capacity of 140 pounds.What is the maximum number of melons that should be placed in a single packing containerif the nominal weight limit is to be exceeded only 5% of the time? Give reasons for youranswer.The manager of a construction job needs to figure prices carefully before submitting a bid. Healso needs to account for uncertainty (variability) in the amounts of products he might need.To oversimplify the real situation, suppose that a project manager treats the amount of sand, inyards, needed for a construction project as a random variable Y₁, which is normally distributedwith mean 10 yards and standard deviation .5 yard. The amount of cement mix needed, inhundreds of pounds, is a random variable Y2, which is normally distributed with mean 4 andstandard deviation .2. The sand costs $7 per yard, and the cement mix costs $3 per hundredpounds. Adding $100 for other costs, he computes his total cost to beU = 100+7Y₁ +3Y₂.If Y₁ and Y₂ are independent, how much should the manager bid to ensure that the true costswill exceed the amount bid with a probability of only .01? Is the independence assumptionreasonable here?Suppose that Y has a gamma distribution with a = n/2 for some positive integer n and ßequal to some specified value. Use the method of moment-generating functions to show thatW = 2Y/B has a x² distribution with n degrees of freedom.A random variable Y has a gamma distribution with a = 3.5 and ß = 4.2. Use the result inExercise 6.46 and the percentage points for the x² distributions given in Table 6, Appendix 3,to find P(Y 33.627).6.48 In a missile-testing program, one random variable of interest is the distance between the pointat which the missile lands and the center of the target at which the missile was aimed. If wethink of the center of the target as the origin of a coordinate system, we can let Y₁ denote

Question

Please do number 6.46 and show work for finding the moment generating function of6.406.416.426.43*6.446.456.466.47function of U = Y² + Y²/².Suppose that Y₁ and Y2 are independent, standard normal random variables. Find the densityLet Y₁, Y2, ..., Y, be independent, normal random variables, each with mean and variance o².U = Σi=1 a¡Y₁.Let a₁, a2, ..., an denote known constants. Find the density function of the linear combinationA type of elevator has a maximum weight capacity Y₁, which is normally distributed with mean5000 pounds and standard deviation 300 pounds. For a certain building equipped with this typeof elevator, the elevator's load, Y2, is a normally distributed random variable with mean 4000pounds and standard deviation 400 pounds. For any given time that the elevator is in use, findthe probability that it will be overloaded, assuming that Y₁ and Y₂ are independent.mean and variance o2.Refer to Exercise 6.41. Let Y₁, Y₂, ..., Y,, be independent, normal random variables, each withaFind the density function of Y =Exercises 323-=nnΣΥi=1Y₁.16 and n =b If o²25, what is the probability that the sample mean, Y, takes on a valuethat is within one unit of the population mean, u? That is, find P(Y- | ≤ 1).c If o² = 16, find P(Y - μ| ≤ 1) if n = 36, n = 64, and n = 81. Interpret the results ofyour calculations.The weight (in pounds) of &#34;medium-size&#34; watermelons is normally distributed with mean 15and variance 4. A packing container for several melons has a nominal capacity of 140 pounds.What is the maximum number of melons that should be placed in a single packing containerif the nominal weight limit is to be exceeded only 5% of the time? Give reasons for youranswer.The manager of a construction job needs to figure prices carefully before submitting a bid. Healso needs to account for uncertainty (variability) in the amounts of products he might need.To oversimplify the real situation, suppose that a project manager treats the amount of sand, inyards, needed for a construction project as a random variable Y₁, which is normally distributedwith mean 10 yards and standard deviation .5 yard. The amount of cement mix needed, inhundreds of pounds, is a random variable Y2, which is normally distributed with mean 4 andstandard deviation .2. The sand costs $7 per yard, and the cement mix costs $3 per hundredpounds. Adding $100 for other costs, he computes his total cost to beU = 100+7Y₁ +3Y₂.If Y₁ and Y₂ are independent, how much should the manager bid to ensure that the true costswill exceed the amount bid with a probability of only .01? Is the independence assumptionreasonable here?Suppose that Y has a gamma distribution with a = n/2 for some positive integer n and ßequal to some specified value. Use the method of moment-generating functions to show thatW = 2Y/B has a x² distribution with n degrees of freedom.A random variable Y has a gamma distribution with a = 3.5 and ß = 4.2. Use the result inExercise 6.46 and the percentage points for the x² distributions given in Table 6, Appendix 3,to find P(Y> 33.627).6.48 In a missile-testing program, one random variable of interest is the distance between the pointat which the missile lands and the center of the target at which the missile was aimed. If wethink of the center of the target as the origin of a coordinate system, we can let Y₁ denote

Accepted Answer

Let Y has gamma distribution with α=n/2 Define W=2Y/β has χ2 with n degree of freedom

n/2 for some positive integer n and B 6.46 Suppose that Y has a gamma distribution with a = equal to some specified value. Use the method of moment-generating functions to show that W = 2Y/B has a x² distribution with n degrees of freedom. nich 35 and 6 42 Use the result in