Let Z be a continuous RV with strictly increasing CDF Fz, and let X = µ+oZ for some real numbers u, o, with o > 0. (i) Let Fx be the CDF of X. Show that Fx(t) = Fz (¹−¹). (ii) Suppose Z~ N(0, 1). Use the previous part to show X ~. ·N(µ‚0²). (iii) Let qz and qx be the quantile functions of Z, X. Show that qx(p) = µ+oqz(p). (Hint: use the relation Q(p) = t⇒ F(t) = p. ) (iv) Below are 15 randomly generated numbers from a normal distribution. 12.1 10.3 9.1 9.9 14 12.6 7 9.6 9.4 11.2 10.6 9.8 10.1 10.8 9.8 Draw the quantile-quantile plot (Q-Q plot) of the sample percentiles p against the quantiles qz (p) of the standard normal to verify it is close to a straight line πp=b+kqz(p). but the slope k 1 and the vertical intercept b 0. What are their values instead? (v) Use the results from parts (ii) and (iii) to infer that the sample in part (iv) does not come from a standard normal distribution N(0, 1), but comes from a normal distribu- tion N(b, k²).

Let Z be a continuous RV with strictly increasing CDF Fz, and let X = µ+oZ for some real numbers u, o, with o > 0. (i) Let Fx be the CDF of X. Show that Fx(t) = Fz (¹−¹). (ii) Suppose Z~ N(0, 1). Use the previous part to show X ~. ·N(µ‚0²). (iii) Let qz and qx be the quantile functions of Z, X. Show that qx(p) = µ+oqz(p). (Hint: use the relation Q(p) = t⇒ F(t) = p. ) (iv) Below are 15 randomly generated numbers from a normal distribution. 12.1 10.3 9.1 9.9 14 12.6 7 9.6 9.4 11.2 10.6 9.8 10.1 10.8 9.8 Draw the quantile-quantile plot (Q-Q plot) of the sample percentiles p against the quantiles qz (p) of the standard normal to verify it is close to a straight line πp=b+kqz(p). but the slope k 1 and the vertical intercept b 0. What are their values instead? (v) Use the results from parts (ii) and (iii) to infer that the sample in part (iv) does not come from a standard normal distribution N(0, 1), but comes from a normal distribu- tion N(b, k²).

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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