for
Section 3: Mean, Median, Variance and Standard Deviation
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35-40. Proofs
41. $0.35$ 43. $0.61$ 45. $0.65$ 47. $0.41$ 49. $0.83$ 51. $E(X^2) = 3, E(X^2) - E(X)^2 = 3/4$ 53. $E(X^2) = 50, E(X^2) - E(X)^2 = 50/9$ 55. $E(X^2) = 1/5, E(X^2) - E(X)^2 = 0.059375$ 57. $E(X^2) = 0.1883, E(X^2) - E(X)^2 = 0.0391$ 59. $E(X^2) = 200, E(X^2) - E(X)^2 = 100$ 61. Comparing answers suggests that $E(X^2) - E(X)^2 = Var(X).$ Thus, $E(X^2) = E(X)^2 + Var(X).$ 63. $$25,000$ 65. $34$ months 67. $8,267$ years 69. $100,000,000$ years 71. $5.3009$ 73. $100.4988$ 75 (a) $β = 2.5556, f(x) = 16.1975x^{2.5556}(1-x)$ 75 (b) $M(X) = 0.66,$ a little smaller than the mean. This indicates that more students scored below the mean than above it.77.
79. Missing words: variance, median.
81. Values of $X$ are more likely to be close to the mode than anywhere else. Thus an interval about the mode determines the most popular values of $X.$
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Stefan Waner (matszw@hofstra.edu) | Steven R. Costenoble (matsrc@hofstra.edu) |