1. BIG Corporation produces just about everything but is currently interested in the lifetimes of its batteries, hoping to obtain its share of a market boosted by the popularity of portable CD and MP3 players. To investigate its new line of Ultra batteries, BIG randomly selects 1000 Ultra batteries and finds that they have a mean lifetime of 860 hours, with a standard deviation of 86 hours. Suppose that this mean and standard deviation apply to the population of all Ultra batteries Complete the following statements about the distribution of lifetimes of all Ultra batteries. (a) According to Chebyshev's theorem, at least (Choose one) lifetimes lie between 688 hours and 1032 hours. b) According to Chebyshev's theorem, at least 36% of the lifetimes lie between hours and hours. (Round your answer to the nearest whole number. ) 2. Students at a major university are complaining of a serious housing crunch. Many of the university's students, they complain, have to commute too far to school because there is not ng near campus. The university officials respond with the following information: the mean distance commuted to school by of the distance commuted is 3.2 miles. Assuming that the university officials' inf statements about the distribution of commute distances for students at this university. (a) According to Chebyshev's theorem, at least o (about 89%) of the commute distances lie between miles and miles. (Round your answer to 1 decimal place.) b) According to Chebyshev's theorem, at least (Choose one) V of the commute distances lie between 11.2 miles and 24.0 miles. . Loretta, who turns eighty this year, has just learned how her blood pressure compares to those of her peers. Specifically, she is interested in her systolic blood pressure, which can be problematic among the elderly. She has uncovered an article in a scientific journal that reports that the mean systolic blood pressure measurement for women venty-five is 131.0 mmHg, with a standard deviation of 6.2 mmHg. Assume that the article reported correct information. Complete the following statements about the distribution of systolic blood pressure measurements for women over seventy-five. (a) According to Chebyshev's theorem, at least (Choose one) V of the measurements lie between 118.6 mmHg and 143.4 mmHg. (b) According to Chebyshev's theorem, at least 84% of the measurements lie between Immig and I mmHg. (Round your answer to 1 decimal place.) 4. Fill in the P (X= x) values to give a legitimate probability distribution for the discrete andom variable .X, whose possible values are - 1, 1, 4, 5, and 6. Value x of X P(X = x) 0.10 0.30 0 5. Fill in the P (Y= x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are -2, 1, 2, 4, and 5. Value x of X P(X = x) -2 0.17 0.23 0 0.25 6. Fill in the P(X= x) values to give a legitimate probability distribution for the Value x of X P (X - x) 0.22 0.21 0.24 Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (g) and "boys" (b), which we write gbg, bbb, etc. For each outcome, let R be the random variable counting the number of girls in each outcome. For example, if the outcome is ggg, then R (ggg) = 3. Suppose that the random variable _X' is defined in terms of R as follows: X = 2R* -4R-2. The values of X" are given in the table below. Outcome bbg gbg ggg bgb bbb gbb bgg ggb Value of X - 4 - 2 4 - 4 - 2 - 4 - 2 - 2 Calculate the values of the probability distribution function of X, i.e. the function Px- First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value x or X D 0 0 Px (x) 0 0 0 . Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (8) and "boys" (6), which we write gbe. bbb, etc. For each outcome, let R be the random variable counting the number of boys in each outcome. For example, if the outcome is gbg, then R(gbg) = 1. Suppose that the random variable X is defined in terms of R as follows: X - 4R-2" -2. The values of X are given in the table below. Outcome bg8 828 bbg 886 8bg bgb gbb bbb 0 -2 -2 0 0 - 2 - 2 - 8 Calculate the values of the probability distribution function of X, i.e. the function py. First, fill in the first row ith the values of X. Then fill in the appropr robabilities in the second row. Value x of X 0 0 0 Px (X) 0 0 0 9. Suppose that the genders of the three children of a certain family are soon to be revealed. Outcomes are thus triples of "girls" (g) and "boys" (b), which we write gbg, bbb, etc. For each outcome, let R be the random variable counting the number of boys in each outcome. For example e, if the outcome is gob, then R(gbb) = 2. Suppose that the random variable X is defined n terms of R as follows: X = R-2. The values of X are given in the table below. Outcome bbg 828 bb bbb gbg bgg ggb gbb Value of X 0 - 2 0 1 - 1 - 1 - 1 0 Calculate the values of the First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row. Value Xor X D 0 0 0 Px (x) 0 0 0 0 10. Let ) be a random variable with the following probability distribution. Value x of X P(X-x) 10 0.30 20 0.35 30 0.20 40 0.15 Complete the following. (If necessary, consult a list of formulas.) a) Find the expectation E(X') of X E(X) = 0 ) Find the variance Var(X') of X. Var (X) = 0