EBook Problems MultivariateNormal - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T18:02:42Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-25T01:17:38Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-25T01:04:52Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-25T00:44:13Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-23T03:30:39Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-21T10:16:02Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-15T23:11:52Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-14T07:03:47Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-14T03:05:07Z

‎EBook Problems Set - Mutivariate Normal Distribution

JayZzz: /* EBook Problems Set - Mutivariate Normal Distribution */

2011-01-14T01:14:58Z

‎EBook Problems Set - Mutivariate Normal Distribution

← Older revision		Revision as of 18:02, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php/EBook_Problems_MultivariateNormal}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php/EBook_Problems_MultivariateNormal}}

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 ''(b) A number of piles may be tied together to form a pile group to resist an external load. Suppose the capacity of the pile group is the sum of the capacities of the individual piles. Consider a pile group that consists of two single piles as described above. Because of proximity, the capacities between the two piles are correlated with a correlation coefficient of 0.8. Let T denote the capacity of this pile group.
-(i) Determine the mean value and cov of T.
+''(i) Determine the mean value and cov of T.
-(ii) Will the random variable T follow a normal, or lognormal or some other probability distribution? Please substantiate.
+''(ii) Will the random variable T follow a normal, or lognormal or some other probability distribution? Please substantiate.
 ===Problem 11===

@@ Line 438: / Line 438: @@
 ''(a) Determine the mean value and standard deviation of the total rate of inflow to pipe 5.
-''(b) What is the probability that during a ''one minute interval'', the volume of water flowing into pipe 2 exceeds that into pipe 1 by at least 400 cubic feet? (Hint: assume the inflow rate into each pipe is constant during that minute)
+''(b) What is the probability that during a '''one minute interval''', the volume of water flowing into pipe 2 exceeds that into pipe 1 by at least 400 cubic feet? (Hint: assume the inflow rate into each pipe is constant during that minute)
 ''(c) Suppose the municipal waste water is projected to increase at a rate of 3 cfs per year, and pipe 5 has a capacity of 70 cfs. If the design criterion is that the probability of overflow at pipe 5 (total inflow exceeds capacity) after a storm should be less than 0.05, how many years will the current pipe 5 remain adequate, i.e., before a larger size pipe is needed?
 <hr>
 * [[EBook | Back to Ebook]]

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 ''(b) What is the probability that the magnitude of the differential settlement (i.e., the difference between the settlements of two adjacent footings) will be less than 0.5 inch?
 <hr>
 * [[EBook | Back to Ebook]]

← Older revision		Revision as of 03:30, 23 January 2011
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	Therefore, extending the channel capacity to about <math>104.5</math> <math>m^3/sec</math> will cut the risk by half.		Therefore, extending the channel capacity to about <math>104.5</math> <math>m^3/sec</math> will cut the risk by half.
	}}		}}
		+
		+	===Problem 5===
		+	Fibers may be embedded in cement to increase its strength. Consider a cracked section as shown in the following diagram:
		+
		+	[[Image:Problem 5.jpg\|250x250px\|center]]
		+
		+	Suppose the total strength T of the cracked section is given by the following expression:
		+
		+	<math>
		+	T=C+(F_1+F_2+...+F_N)
		+	</math>
		+
		+	Where C is the strength contributed by the cement; <math>F_1</math>, <math>F_2</math>, etc. are the strength of each of the fibers across the crack; N is the total number of fibers across the crack. Suppose C=N(30,5); each <math>F_i</math> is N(5,3) and N is a discrete random variable with the following PMF
		+
		+
		+	[[Image:Problem 5 PMF.jpg\|600x600px\|center]]
		+
		+	Assume C and <math>F_i</math>'s are statistically independent. What is the probability that the total strength T will be less than 30?
		+
		+	===Problem 6===
		+	[[Image:Problem 6.jpg\|300x300px\|center]]
		+
		+	The settlement of each footing shown follows a normal distribution with a mean of 2 inches and a coefficient of variation of 30%. Suppose the settlements between two adjacent footings are correlated with a correlation coefficient of 0.7. suppose
		+
		+	<math>D=S_1-S_2</math>
		+
		+	where <math>S_1</math> and <math>S_2</math> are settlements of footings 1 and 2, respectively.
		+
		+	''(a) Determine the mean and variance of D.
		+
		+	''(b) What is the probability that the magnitude of the differential settlement (i.e., the difference between the settlements of two adjacent footings) will be less than 0.5 inch?

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 \cong 96 years \,
 </math>
 }}
 <hr>
 * [[EBook | Back to Ebook]]

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 \Rightarrow\sigma_x=\sqrt{300} \cong 17.32(m^3/sec) \,
 </math>
 }}

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 ''(d) If the desired flooding risk over a 10 year period is to be reduced by half, how large should the present channel capacity be extended to?
 <hr>
 * [[EBook | Back to Ebook]]

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 \cong0.00114 \,
 </math>
-}}
+''(a) What are the mean and standard deviation of the annual maximum peak discharge passing through the city?
 <hr>

← Older revision		Revision as of 01:14, 14 January 2011
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	}}		}}

		+	===Problem 3===
		+	A friction pile is driven through three soil layers as shown in the following figure:

		+	[[Image:Problem 3.jpg\|250x250px\|center]]

		+	The total bearing capacity of the friction pile (in tons) is obtained from
		+
		+	<math>
		+	Q=4A+B+2C
		+	</math>
		+
		+	Where A, B, and C are penetration lengths (in meters) through each of the three soil layers, respectively. Suppose A ~ N(5,3); B ~ N(8,2); A and B are negatively correlated with coefficient <math>\rho</math> = - 0.5. The total length of the pile is 30 meters. Determine the probability that the pile will fail to support the 40 ton load, i.e., the event that the capacity Q is less than 40 tons. [Hint: Observe C = 30 - (A+B)].
		+
		+	{{hidden\|Answer\|
		+	<math>
		+	Q=4A+B+2C \,
		+	</math>
		+
		+	<math>
		+	=4A+B+2(30-A-B) \,
		+	</math>
		+
		+	<math>
		+	=2A-B+60 \,
		+	</math>
		+
		+	Hence,
		+
		+	<math>
		+	\mu_Q=2 x 5-8+60=62 \,
		+	</math>
		+
		+	<math>
		+	\sigma_Q=[4 x 3^2+1 x 2^2 +2(2)(-1)(-0.5)(3)(2)]^{1/2} \,
		+	</math>
		+
		+	<math>
		+	=(36+4+12)^{1/2}=\sqrt{52} \,
		+	</math>
		+
		+	Hence
		+
		+	<math>
		+	P(Q<40)=P(\frac{Q-\mu_Q}{\sigma_Q}<\frac{40-62}{\sqrt{52}}) \,
		+	</math>
		+
		+	<math>
		+	=\Phi(-3.050851079) \,
		+	</math>
		+
		+	<math>
		+	\cong0.00114 \,
		+	</math>
		+
		+	}}

	<hr>		<hr>