 ## Calculus Applied to Probability and Statistics by Stefan Waner and Steven R. Costenoble ## ExercisesforSection 3: Mean, Median, Variance and Standard Deviation 2. Probability Density Functions: Uniform, Exponential, Normal, and Beta 3. Mean, Median, Variance and Standard Deviation 4. You're the Expert Creating a Family Trust Calculus and Probability Main Page "Real World" Page Answers to Odd-Numbered Exercises Find the expected value \$E(X),\$ the variance \$Var(X)\$ and the standard deviation \$β(X)\$ for each of the density functions in Exercises 1 through 20.

 1. \$f(x) = 1/3\$  on \$[0,  3]\$ 2. \$f(x) = 3\$  on \$[0,  1/3]\$ 3. \$f(x) = x/50\$  on \$[0,  10]\$ 4. \$f(x) = 5x\$  on \$[0,  (2/5)^{1/2}]\$ 5. \$f(x) = (3/2) (1 - x^2)\$  on \$[0,  1]\$     6. \$f(x) = (3/4)(1 - x^2)\$  on \$[-1,  1]\$ 7. \$f(x) = e^x\$  on \$[0,  ln 2]\$ 8. \$f(x) = 1/x\$  on \$[0,  e]\$ 9. \$f(x) = 0.1e^{-0.1x}\$ on \$[0,  +∞)\$ 10. \$f(x) = 4e^{4x}\$ on \$(-∞,x 0]\$ 11. \$f(x) = 0.03e6{0.03x}\$  on \$(-∞,  0]\$ 12. \$f(x) = 0.02e^{-0.02x}\$  on \$[0,  +∞)\$ 13. \$f(x) = 2/x^3\$  on \$[1,  +∞)\$ 14. \$f(x) = 1/2x^{0.5}\$  on \$(0,  1]\$ 15. Normal density function with \$µ = 1\$ and \$σ = 1\$ on \$(-∞,  +∞)\$ 16. Normal density function with \$µ = -1\$ and \$σ = 1\$ on \$(-∞,  +∞)\$ 17. Beta density function with \$β = 0.5\$ 18. Beta density function with \$β = 1.5\$ 19. Beta density function with \$β = 3.2\$ 20. Beta density function with \$β = 4.6\$

Use a graphing calculator or computer to find \$E(X),\$  \$Var(X)\$ and \$σ(X)\$ for each of the density functions in Exercises 21 through 24. (Round all answers to four significant digits.) 21. \$f(x) = 4/[π(1+x^2)]\$ on \$[0,  1]\$ 22. \$f(x) = 3/[π(1-x^2)^{1/2}]\$ on \$[1/2,  1]\$ 23. \$f(x) = 2xe^{-x^2}\$ on \$[0,  +∞)\$ 24. \$f(x) = -2xe^{-x^2}\$ on \$(-∞,  0]\$

Find the medians of the random variables with the probability density functions given in Exercises 25 through 34.

 25. \$f(x) = 0.25\$ on \$[0,  4]\$ 26. \$f(x) = 4\$ on \$[0,  0.25]\$ 27. \$f(x) = 3e^{-3x}\$ on \$[0,  +∞)\$ 28. \$f(x) = 0.5e^{-0.5x}\$ on \$[0,  +∞)\$ 29. \$f(x) = 0.03e^{0.03x}\$ on \$(-∞,  0]\$ 30. \$f(x) = 0.02e^{-0.02x}\$ on \$[0,  +∞)\$ 31. \$f(x) = 2(1 - x)\$ on \$[0,  1]\$ 32. \$f(x) =\$ on \$[1,  +∞)\$ 33. \$f(x) = 1/(2x^{1/2})\$ on \$(0,  1]\$ 34. \$f(x) = 1/x\$ on \$[1,  e]\$

35. Mean of a Uniform Distribution Verify the formula for the mean of a uniform distribution by computing the integral.

36. Mean of a Beta Distribution Verify the formula for the mean of a beta distribution by computing the integral.

37. Variance of a Uniform Distribution Verify the formula for the variance of a uniform distribution by computing the integral.

38. Variance of an Exponential Distribution Verify the formula for the variance of an exponential distribution by computing the integral.

39. Median of an Exponential Random Variable Show that if \$X\$ is a random variable with density function \$f(x) = ae^{-ax}\$ on \$[0,  +∞),\$ then \$X\$ has median \$ ^{ln 2}/a.\$

40. Median of a Uniform Random Variable Show that if \$X\$ is uniform random variable taking values in the interval \$[a,  b],\$ then \$X\$ has median \$(a+b)/2.\$

Use a graphing calculator or computer to find the medians of the random variables with the probability density functions given in Exercises 41 through 50. (Round all answers to two decimal places.) 41. \$f(x) = (3/2)(1 - x^2)\$ on \$[0,  1]\$ 42. f(x) = (3/4)(1 - x2) on [-1, 1] 43. beta density function with \$β = 2\$ 44. beta density function with \$β = 3\$ 45. beta density function with \$β = 2.5\$ 46. beta density function with \$β = 0.5\$ 47. \$f(x) = 4/[π(1+x^2)]\$ on \$[0,  1]\$ 48. \$f(x) = 3/(π(1-x^2)^{1/2})\$ on \$[1/2,  1]\$ 49. \$2xe^{-x^2}\$ on \$[0,  +∞)\$ 50. \$f(x) = - 2xe^{-x^2}\$ on \$(-∞,  0]\$

The mean square of a random variable \$X\$ with density function \$f\$ is given by the formula
\$E(X^2) =\$ ∫ab\$x^2 f(x) dx.\$

51-60. In Exercises 1-10, compute \$E(X^2).\$ In each case, compute also \$E(X^2) - E(X)^2.\$

61. Compare the answers in 51-60 to those in 1-10, and hence suggest a formula expressing \$E(X^2)\$ in terms of \$E(X)\$ and \$Var(X).\$

62. Calculate \$E(e^{tX}) =\$ ∫abetx\$f(x) dx\$  with \$f(x) = 0.1e^{-0.1x}\$ on \$[0,  +∞).\$ Then evaluate \$(d/dt)E^{tX}|_{t=0}\$ and \$(d^2/dt^2])E^{tX}|_{t=0},\$ comparing these answers with the answer to Exercise 60. What do you notice?

Applications

63. Salaries Assuming that workers' salaries in your company are uniformly distributed between \$\$10,000\$ and \$\$40,000\$ per year, calculate the average salary in your company.

64. Grades The grade point averages (gpa's) of members of the Gourmet Society are uniformly distributed between \$2.5\$ and \$3.5.\$ Find the average gpa in the Gourmet Society.

65. Boring Television Series Your company's new series "Avocado Comedy Hour" has been a complete flop, with viewership continuously declining at a rate of 30% per month. How long will the average viewer continue to watch the show?

66. Bad Investments Investments in junk bonds are declining continuously at a rate of \$5%\$ per year. How long will an average dollar remain invested in junk bonds?

67. Radioactive Decay The half-life of carbon-\$14\$ is \$5,730\$ years. How long, to the nearest year, do you expect it to take for a randomly selected carbon-\$14\$ atom to decay?

68. Radioactive Decay The half-life of plutonium-\$239\$ is \$24,400\$ years. How long, to the nearest year, do you expect it to take for a randomly selected plutonium-\$239\$ atom to decay?

69. The Doomsday Meteor The probability that a "doomsday meteor" will hit the earth in any given year and release a billion megatons or more of energy is on the order of \$0.000 000 01.^*\$ When do you expect the earth to be hit by a doomsday meteor? (Use an exponential distribution with \$a = 0.000 000 01.\$)

70. Galactic Cataclysm The probability that the galaxy \$MX-47\$ will explode within the next million years is estimated to be \$0.0003.\$ When do you expect \$MX-47\$ to explode? (Use an exponential distribution with \$a = 0.0003.\$)

Exercises 71-74 use the normal probability density function and require the use of technology for numerical integration. (Alternatively, see Exercise 61.) Find the root mean square value for \$X (i.e., (E(X^2))^{1/2})\$ in each exercise. 71. Physical Measurements Repeated measurements of a metal rod yield a mean of \$5.3\$ inches, with a standard deviation of \$0.1.\$ 72. IQ Testing Repeated measurements of a student's IQ yield a mean of \$135,\$ with a standard deviation of \$5.\$ 73. Psychology Tests It is known that subjects score an average of 100 points on a new personality test, with a standard deviation of \$10\$ points. 74. Examination Scores Professor May's students earned an average grade of \$3.5\$ with a standard deviation of \$0.2.\$

75. Learning A graduate psychology student finds that \$64%\$ of all first semester calculus students in Prof. Mean's class have a working knowledge of the derivative by the end of the semester.

(a) Take \$X =\$ percentage of students who have a working knowledge of calculus after \$1\$ semester, and find a beta density function that models \$X,\$ assuming that the performance of students in Prof. Mean's is average.

(b) Find the median of \$X\$ (rounded to two decimal places) and comment on any difference between the median and the mean.

76. Plant Shutdowns An automobile plant is open an average of \$78%\$ of the year.

(a) Take \$X =\$ fraction of the year for which the plant is open, and find a beta density function that models \$X.\$

(b) Find the median of \$X\$ (rounded to two decimal places) and comment on any difference between the median and the mean.

Communication and Reasoning Exercises

77. Sketch the graph of a probability distribution function with the property that its median is smaller than its mean.

78. Sketch the graph of a probability distribution function \$(0  ≤  X  ≤  1)\$ with a large standard deviation and a small mean.

79. Complete the following sentence. The ___ measures the degree to which the values of \$X\$ are distributed, while the ___ is the value of \$X\$ such that half the measurements of \$X\$ are below and half are above (for a large number of measurements).

80. Complete the following sentence. (See Exercise 61.) Given two of the quantities ___, ___ and ___, we can calculate the third using the formula ___ .

81. A value of \$X\$ for which the probability distribution function \$f\$ has a local maximum is called a mode of the distribution. (If there is more than one mode, the distribution is called bimodal (2 modes), trimodal (3 modes), etc. as the case may be.) If a distribution has a single mode, what does it tell you?

82. Referring to Exercise 81, sketch a bimodal distribution whose mean coincides with neither of the modes.  2. Probability Density Functions: Uniform, Exponential, Normal, and Beta 3. Mean, Median, Variance and Standard Deviation 4. You're the Expert Creating a Family Trust Calculus and Probability Main Page "Real World" Page Answers to Odd-Numbered Exercises We would welcome comments and suggestions for improving this resource. Mail us at: Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: September, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble
* Source: NASA International Near-Earth-Object Detection Workshop (New York Times, January 25, 1994, p. C1.)