Difference between revisions of "EBook Problems MultivariateNormal"
(→EBook Problems Set - Mutivariate Normal Distribution) |
(→EBook Problems Set - Mutivariate Normal Distribution) |
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| − | ''(a) What is the probability that Person2 will not arrive at point D within 10 hours? | + | ''(a) What is the probability that Person2 will not arrive at point D within 10 hours? |
| − | ''(b) | + | ''(b) What is the probability that Person1 will arrive at point D earlier than Person2 by at least one hour? |
| − | ''(c) | + | ''(c) Which route (A<math>\rightarrow</math>B<math>\rightarrow</math>D or A<math>\rightarrow</math>C<math>\rightarrow</math>D) should be taken if one wishes to minimize the expected travel time from A to D? Explain. |
| − | + | {{hidden|Answer for part (a)| | |
| − | {{hidden|Answer for part (a)| | + | |
| − | Let T2 be Person2’s travel time in (hours); T2 = | + | Let ''T2'' be Person2’s travel time in (hours); |
| + | <math> | ||
| + | T2 = T_3 + T_4 | ||
| + | </math> | ||
| + | with | ||
| + | <math> | ||
| + | \mu_{T2}=5+4=9(hours), | ||
| + | </math> | ||
| + | and | ||
| + | |||
| + | <math>\sigma_{T2}=[3^2+1^2+(2)(0.8)(3)(1)]^{1/2}=3.847076812 (hours)</math> | ||
| + | |||
| + | Hence | ||
| + | |||
| + | <math> | ||
| + | P(T2>10hours)=1-P(\frac{T2-\mu_{T2}}{\sigma_{T2}}\leq\frac{10-9}{3.847076812}) | ||
| + | |||
| + | =1-\Phi(0.259937622)=1-0.602543999 | ||
| + | |||
| + | \cong 0.397 | ||
| + | </math> | ||
}} | }} | ||
| + | {{hidden|Answer for part (b)| | ||
| + | Let ''T1'' be Person1's travel time in (hours); with | ||
| + | |||
| + | <math> | ||
| + | \mu_{T1}=6+4=10(hours), | ||
| + | </math> | ||
| + | and | ||
| + | |||
| + | <math> | ||
| + | \sigma_{T1}=[2^2+1^2]^{1/2}=\sqrt{5} | ||
| + | </math> | ||
| + | |||
| + | Hence | ||
| + | |||
| + | <math> | ||
| + | P(T2-T1>1)=P(T1-T2+1<0), | ||
| + | </math> | ||
| + | Now let R <math>\equiv</math> T1-T2+1; R is normal with | ||
| + | <math> | ||
| + | \mu_R=\mu_{T1}-\mu_{T2}+1=10-9+1=2, | ||
| + | </math> | ||
| + | |||
| + | <math> | ||
| + | \sigma_R=[\sigma_{T1}^2+\sigma_{T2}^2]^{1/2}=(5+14.8)^{1/2}=\sqrt{19.8}, | ||
| + | </math> | ||
| + | |||
| + | Hence | ||
| + | |||
| + | <math> | ||
| + | P(R<0)=P(\frac{R-\mu_R}{\sigma_R}<\frac{0-2}{\sqrt{19.8}}) | ||
| + | </math> | ||
| + | |||
| + | Hence | ||
| + | |||
| + | <math> | ||
| + | =\Phi (-0.44946657) | ||
| + | </math> | ||
| + | |||
| + | <math> | ||
| + | \cong 0.327 | ||
| + | </math> | ||
| + | }} | ||
| + | {{hidden| Answer for part (c) | | ||
| + | Since the A<math>\rightarrow</math>C<math>\rightarrow</math>D route has a smaller expected travel time of <math>\mu_{T2}=9</math> hours as compared to the upper (with expected travel time <math>=\mu_{T1}=10 hours)</math>, one should take the '''lower''' route to minimized expected travel time from A to D. | ||
| + | }} | ||
<hr> | <hr> | ||
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Revision as of 21:25, 6 January 2011
EBook Problems Set - Mutivariate Normal Distribution
Problem 1
Person1 and Person2 are travelling from point A to point D, but there are different routes to get from A to D. Person1 decides to take the A->B->D route, whereas Person2 takes the A->C->D route.
The travel times (in hours) between each pair of points indicated are normally distributed as follows:
T1 ~ N (6, 2)
T2 ~ N (4, 1)
T3 ~ N (5, 3)
T4 ~ N (4, 1)
Explain why these times are stochastic (and not exact or deterministic)? Although the travel times here generally can be assumed statistically independent, T3 and T4 are dependent with correlation coefficient 0.8.
(a) What is the probability that Person2 will not arrive at point D within 10 hours?
(b) What is the probability that Person1 will arrive at point D earlier than Person2 by at least one hour?
(c) Which route (A\(\rightarrow\)B\(\rightarrow\)D or A\(\rightarrow\)C\(\rightarrow\)D) should be taken if one wishes to minimize the expected travel time from A to D? Explain.
{\sigma_{T2}}\leq\frac{10-9}{3.847076812})
=1-\Phi(0.259937622)=1-0.602543999
\cong 0.397 \) }}
) \)
Hence
\( =\Phi (-0.44946657) \)
\( \cong 0.327 \) }}
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