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8. For each of the following calculate the quantity indicated. [4 points each) (a) Suppose that...
Problem 4 (5 points) a. (2 points) Suppose X ~ Gam(2,0.5). Compute the probability P(4 < X < 6). Explain. b. (3 points) Suppose X ~ Gam(a, 1), where a > 0 and 1 > 0. Determine E[X]. Use this result to compute the expected value of a Gam(2,0.5) random vari- able. Explain (you may want to use the probability density function (PDF) of a Gam(a +1,ß) random variable to compute E[X]).
4. (8 Marks) Suppose X is a random variable best described by a uniformly distribution or probability that ranges from 2 to 11. a) Write down the probability density function f(1). (1.5 points) b) Compute the following: i) mean (1.5 points) ii) standard deviation (1.5 points) iii) P(X < 3.858) (1.5 points) iv) P(-O< X <H+ o) (2 points)
a) In each of the following pmfs, find the value of C. i) p(x) Cx, x 1, 2, 3, 4, 5 ii) p(x) C/x, 2,4,8, 16 b) Assume that the pmf of a discrete random variable X is given by px (x) = 20x-, x = 1, 2, 3, Calculate the following probabilities: i) P(X <3) ii) E[X]
(e) Suppose that 4 balls are placed sequentially into one of 5 bins, where the bin for each ball is selected at random. For i = 1, 2,3,4,5, define the indicator variables 1, if the ith bin is empty; 0, otherwise Xi _ Then the number of empty boxes is given by X = X\+X2+ X3 + X4+X5, and we learned from week #7 lecture notes and midterm II that Xi ~ Bernoulli(p) with p = (1 - 2)4 =...
8. Suppose that the joint density of X and Y is given by e if 0< I<, 0<y< 0, otherwise. Find P(X > 1 Y = y).
(8 points) Suppose that the number of flaws on an individual tile has the Poisson distribution with param- eter = 1.5. A tile is called non-conforming if it has 2 or more flaws. Suppose that we purchase 6 tiles. What is the probability that 4 or more of the 6 tiles are non-conforming?
6. (8 points) Suppose that the number of flaws on an individual tile has the Poisson distribution with param- eter 1 = 1.5. A tile is called non-conforming if it has 2 or more flaws. Suppose that we purchase 6 tiles. What is the probability that 4 or more of the 6 tiles are non-conforming?
7T Problem  <15 points> -ĉ on a surface given by r = 5—10,0=1,2=4 →8. Calculate the total flux Ø passing through this surface. Magnetic flux density is given as B = 92 6+92 r r
H.W. #5 - Q #8: 8. Suppose that n-48 seeds are planted and suppose that each seed has a probability p 75% of germinating. Let X be the number of seeds that germinate and use the Central Limit Theorem to estimate the probability P(35 < X < 40) that between 35 and 40 seeds germinate. Don't forget to use a continuity correction.
a) (2.5 points) Suppose we have the two following two datasets: (i) 1, 3, 4, 5, 7, 10, 11 (ii) 6, 8, 9, 10, 12, 15, 16 For each dataset, calculate by hand the mean, interquartile range, and the standard deviation. Make sure to show step-by-step calculations. b) (1 point) How is list (ii) related to list (i)? How does this relationship carry over to the average? the standard deviation? ECONOMETRIX COURSE