STAT410- Homework #06 Solved

. Grades on Fall 2021 STAT 410 Exam 1 were not very good *. Graphed, their distribution had a shape similar to the probability density function **

                                                fx(x)80,                                                zero elsewhere.

3, 200

 Recall (Homework #1 ): x +4x-320 (x — 16) (x + 20)

Fx(x)16 SX< 80.

                                       6,400                          6,400

Six exam papers were selected at random. That is, let X , X , X , X , X , X be a random sample ( i.i.d.) of size n = 6 from the above probability distribution.

Let Y < Y < Y < Y < Y < Y be the corresponding order statistics.

g)                  Find the probability that the largest score of these 6 papers is above 68. That is, find

 

h)                  Find the probability that the lowest score of these 6 papers is below 36. That is, find P(YI < 36) = 

i)                    Find the probability that the second lowest score of these 6 papers is above 52. That is, find 

j)                    Find the probability that the third largest score of these 6 papers (the fourth lowest score) is above 60. That is, find P (Y > 60).

 

* The probability distribution is fictional, the exam has not happened yet. Hopefully, the actual grades will be slightly better than these.

 ** Exam scores should have a discrete ( instead of continuous) nature. A continuous probability distribution is used as an approximation, since the alternative would have been dealing with a discrete random variable with 65 possible values ( 16, 17, 18, . 79, 80 ), which is not nearly as much fun as I am describing it here.

5. Mary had a little lamb (we all know the rest). Realizing that she has more love to give, Mary decided to purchase more livestock. At a local market, the prices for a calf (baby cow), C, and a kid (baby goat), K, vary from day to day and jointly follow a bivariate normal distribution with

                                     pc = $434, oc=S10, Pie $222, 0K =$3,                           = 0.60.

a)                  Find the probability that on a given day, a calf costs more than $430. That is, find

 

b)                 Find the probability that on a given day, a kid costs more than $225. That is, find  225).

c)                  Suppose that on a given day, a calf costs $430. Find the probability that a kid costs more than $225. That is, find   lc=430).

d)                 Find the probability that on a given day, a calf costs more than two kids. That is, find

 

e)                  Mary buys 5 calves and 4 kids. Find the probability that she paid more than $3,000.

That is, find 

5.         (continued)

Suppose that the price of a calf, C, the price of a kid K, and the price of a lamb (baby sheep), L, [in dollars ] jointly follow N3(g, E) distribution with

   100 18 24 and18 9 9

                                                                                                24     9    36

f)                  What is the value of p CL ?

g)                Find the probability that on a given day, a lamb costs more than $334. That is, find

 

h)                Mary buys 5 calves, 4 kids, and 3 lambs. Find the probability that she paid more than

$4,000. That is, find