Math 3339 Homework 6 (Sections )

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1 Math 3339 Homework 6 (Sections ) Name: Key PeopleSoft ID: Instructions: Homework will NOT be accepted through or in person. Homework must be submitted through CourseWare BEFORE the deadline. Print out this file and complete the problems. Use blue or black ink or a dark pencil. Write your solutions in the space provided. You must show all work for full credit. Submit this assignment at under Assignments" and choose HW6. Total possible points, 5.. Section 5.2, Problem. For x let f(x) = kx( x), where k is a constant. Find the value of k such that f is a density function. kkkk( xx) dddd = kk xx xx 2 dddd = kk xx2 2 xx3 3 = kk 2 3 = kk 6 = kk = 6

2 2. Section 5.2, Problem 2 Find the mean and variance of the distribution in the preceding exercise. E(X) = 6xx 2 ( xx)dddd = 6 xx 2 xx 3 dddd = 6 xx3 E(X 2 ) = 6xx 3 ( xx)dddd = 6 xx 3 xx 4 dddd = 6 xx4 Var(X) = E(X 2 ) - E(X) 2 = =.3.25 =.5 3 xx4 4 = = 6 2 = 2 4 xx5 5 = = 6 2 = 3 or.5 or.3 Brooks/Cole, 27.

3 3. Find the cumulative distribution for the previous density function. Using the integration from problem, 6 xx2 2 xx3, xx < FF(XX) = xx 2 ( xx), xx, xx > 3 = xx2 ( xx) since 6 is the Least Common Denominator. Brooks/Cole, 27.

4 4. The total number of hours, measured in units of hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function xx, < xx < ff(xx) = 2 xx, xx 2, elsewhere a) Find the probability that over a period of one year, a family runs their vacuum cleaner less than 2 hours. b) Find the probability that over a period of one year, a family runs their vacuum cleaner between 5 and hours. a. P(X <.2) = P( < X < ) + P( X <.2) = xx dddd.2 = xx xx xx2 2 = 2 b. P(.5 < X < ) = xx.5 dddd = xx xx dddd = = = Suppose that a random variable X has a cdf of: xx 2, xx < FF(xx) = 9, xx < 3, xx 3 a) Determine PP(XX ). b) Determine PP(.75 XX.35). c) Determine c for the following probability PP(XX cc) =. d) Determine the median of this probability function. e) Determine the mean, EE(XX). a. P(X ) = F() = or. 9 b. P(.75 X.35) = F(.35) F(.75) =.352 c. P(X c) =. F(c) = cc 2 9 =. cc 2 =.9 cc =.3 d. For the median F(c) =.5 cc 2 9 =.5 cc 2 = 4.5 cc = e. First need to find F (X) = 2xx for x Then E(X) = 2xx2 dddd = 2xx = 2(3)3 = 2 27 =.4 Brooks/Cole, 27.

5 6. * Suppose the reaction temperature X (in C) in a certain chemical process has a uniform distribution by X ~ Unif(-5, 5). a. Give the pdf of X. b. Compute P(X < ). c. Compute P(-2.5 < X < 2.5) d. For k satisfying -5 < k < k +4 < 5, compute P(k < X < k +4). a. ff(xx) =, ooooheeeeeeeeeeee b. PP(XX < ) = or.5 2 c. PP( 2.5 < XX < 2.5) = or.5 2 d. PP(kk < XX < kk + 4) = (kk + 4 kk) = or.4 5 The denominator is the difference from the highest value and lowest value. Brooks/Cole, 27.

6 7. Suppose the time spent by a randomly selected student at a campus computer lab has a gamma distribution with mean 2 minutes and variance 8 minutes 2. a) What are the values of α and β? b) What is the probability that a student spends less than minutes at a campus computer lab? c) What is the probability that a student spends at least hours at a campus computer lab? a. E(X) = 2; Var(X) = 8 E(X) = αβ; Var(X) = αβ 2 2 = αβ α = 2/β 8 = αβ 2 8 = (2/β)β 2 β = 8/2 = 4 α = 2/4 = 5 For finding the probabilities, we will use R-studio and the pgamma function b. P(X < ) = pgamma(,5,/4) =.88 c. P(X 6) = pgamma(6,5,/4) =.86 Brooks/Cole, 27.

7 8. Let X denote the distance an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that X has an exponential distribution with parameter λ =.24. a) What is the probability that the distance is at most 5m? b) What is the expected value of the distance an animal moves from its birth site to the first territorial vacancy it encounters? To find these probabilities we will use R studio and pexp(x,λ). a. P(X 5) = pexp(5,.24) =.9727 b. E(X) = /λ = /.24 = m Brooks/Cole, 27.

8 For questions 9 & circle your best answer. 9. A continuous random variable may assume a. any value in an interval or collection of intervals b. only integer values in an interval or collection of intervals c. only fractional values in an interval or collection of intervals d. only the positive integer values in an interval e. none of these. A description of the distribution of the values of a random variable and their associated probabilities is called a a. probability distribution b. random variance c. random variable d. expected value e. none of these Brooks/Cole, 27.