SOCR EduMaterials Activities Continuous Distributions - Revision history

Rgidwani: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

2007-07-08T19:05:58Z

‎This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson.

Rgidwani: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

2007-07-08T18:59:22Z

‎This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson.

Rgidwani: changed the 30% in exercise 4 from a $to a to keep uniform text.$

2007-07-06T23:04:05Z

changed the 30% in exercise 4 from a <math> to a <no wiki> to keep uniform text.

IvoDinov at 00:33, 29 December 2006

2006-12-29T00:33:55Z

Nchristo: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

2006-11-25T04:48:55Z

‎This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson.

Nchristo: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

2006-11-25T04:47:03Z

‎This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson.

Nchristo at 04:41, 25 November 2006

2006-11-25T04:41:14Z

New page

== This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. ==

* '''Description''': You can access the applets for the above distributions at http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .

*'''Exercise 1:'''
Graph and print <math> X \sim exp(0.2) </math>, <math> X \sim exp(1) </math>, <math> X \sim exp(10) </math>

**'''1.''' Locate the maximum density for each one of these distributions.

**'''2.''' Find the height of the density at 3 values of <math> X </math> (one near 0, one near the mean, and one towards the tail of the distribution).

**'''3.''' Find one percentile for each of these distributions and record them on the printouts. Verify these percentiles using the formula we discussed in class:
<math>
x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}
</math>

**'''4.''' Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:
<math>
P(X \le x)=1-e^{-\lambda x}
</math>

*'''Exercise 2:''' Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>. Show on the graph the following points: <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>. How many standard deviations from the mean is the value <math> x=27.5 </math>?
*'''Exercise 3:''' Graph the distribution of <math> X \sim N(40, 10) </math>.
**'''1.''' Find <math> P(X>49)</math> Submit a printout.
**'''2.''' Find <math>P(X<22)</math> Submit a printout.
**'''3.''' Find <math>P(X<58)</math> Submit a printout.
**'''4.''' Find <math>P(X>13)</math> Submit a printout.
**'''5.''' Find <math>P(12<X<37)</math> Submit a printout.
**'''6.''' Find <math> P(33<X<60)</math> Submit a printout.
**'''7.''' Find <math> P(52<X<65)</math> Submit a printout.
**'''8.''' Use the mouse or the left cut off or right cut off points to find the <math>8_{th}, \ 20_{th}, \ 45_{th}, \ 55_{th}, \ 70_{th}, \ 95_{th}</math> percentiles. After you find these percentiles you can place by hand all of them in one printout (or you can submit a printout for each one of them if you want).
**'''9.''' Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout! You do not need to submit anything here.

*'''Exercise 3:''' The lifetime of tires of brand A follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
**'''1.''' Use SOCR to find the probability that a tire will last between 40000 and 46000 miles.
**'''2.''' Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
**'''3.''' Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.

*'''Exercise 4:''' The probability that a student is admitted in the Math Department Major at a college is <math> 30 \%</math>. Suppose that this year 150 students will apply for admission into the Math major.

**'''1.''' What is the distribution of the number of students admitted? Use SOCR to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
**'''2.''' Write an expression for the exact probability that among the 150 students at least 55 will be admitted.
**'''3.''' Use SOCR to compute the probability of part (2).
**'''4.''' Use the normal distribution to approximate the probability of part (2) (do not forget the continuity correction). What is the error of the approximation?

*'''Exercise 5:''' The number of pine trees per acre in a forest follows the Poisson distribution with mean 30.

**'''1.''' Graph and print this distribution.
**'''2.''' What is the shape of this distribution? Why?
**'''3.''' Use SOCR to find the probability that an acre has more than 37 pines.
**'''4.''' Approximate the probability of part (c) using the normal distribution.

Below you can see the distribution of a normal random variable <math> X </math> with <math> \mu=50, \ \sigma=5 </math>. In this graph you can also see the probability that <math> X </math> is between 53 and 60.

<center>[[Image: SOCR_Activities_Christou_normal.jpg|600px]]</center>

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfidenceIntervals}}

</math>

← Older revision		Revision as of 00:33, 29 December 2006
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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?title=~~SOCR_EduMaterials_Activities_ConfidenceIntervals~~}}	+	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_Continuous_Distributions}}
−
−	~~</math>~~

← Older revision		Revision as of 04:48, 25 November 2006
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	Below you can see the distribution of an exponential random variable <math> X </math> with <math> \lambda=2.5</math>. In this graph you can also see the probability that <math> X </math> is between 0.8 and 1.3.		Below you can see the distribution of an exponential random variable <math> X </math> with <math> \lambda=2.5</math>. In this graph you can also see the probability that <math> X </math> is between 0.8 and 1.3.

−	<center>[[Image: ~~SOCR_Activities_Christou_normal13~~.jpg\|600px]]</center>	+	<center>[[Image: SOCR_Activities_Christou_exp.jpg\|600px]]</center>

@@ Line 4: / Line 4: @@
-*'''Exercise 1:'''
+*'''Exercise 1:''' Graph and print <math> X \sim exp(0.2) </math>, <math> X \sim exp(1) </math>, <math> X \sim exp(10) </math>
 **'''1.'''  Locate the maximum density for each one of these distributions.
 **'''2.''' Find the height of the density at 3 values of <math> X </math> (one near 0, one near the mean, and one towards the tail of the distribution).
 **'''3.''' Find one percentile for each of these distributions and record them on the printouts.  Verify these  percentiles using the formula we discussed in class: <math>x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}</math>
 **'''4.''' Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:
 <math>
@@ Line 44: / Line 36: @@
 *'''Exercise 5:''' The probability that a student is admitted in the Math Department Major at a college is  <nowiki>30%</nowiki>.  Suppose that this year 150 students will apply for admission into the Math major.
 **'''1.''' What is the distribution of the number of students admitted?  Use SOCR to graph and print this distribution.  What is the shape of this distribution?  What is the mean and standard deviation of this distribution?
 **'''2.''' Write an expression for the exact probability that among the 150 students at least 55 will be admitted.
@@ Line 51: / Line 42: @@
 *'''Exercise 6:'''   The number of pine trees per acre in a forest follows the Poisson distribution with mean 30.
 **'''1.''' Graph and print this distribution.
 **'''2.''' What is the shape of this distribution?  Why?

@@ Line 38: / Line 38: @@
 **'''9.'''  Make sure you know how to answer the above questions using the <math>z</math> score <math>z=\frac{x-\mu}{\sigma}</math> and your <math>z</math> table from the handout!  You do not need to submit anything here.
-*'''Exercise 3:''' The lifetime of tires of brand A follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
+*'''Exercise 4:''' The lifetime of tires of brand A follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
 **'''1.'''  Use SOCR to find the probability that a tire will last between 40000 and 46000 miles.
 **'''2.''' Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles?  Submit a printout and explain how you get the answer.
 **'''3.'''  Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles?  Submit a printout and explain how you get the answer.
-*'''Exercise 4:''' The probability that a student is admitted in the Math Department Major at a college is  <nowiki>30%</nowiki>.  Suppose that this year 150 students will apply for admission into the Math major.
+*'''Exercise 5:''' The probability that a student is admitted in the Math Department Major at a college is  <nowiki>30%</nowiki>.  Suppose that this year 150 students will apply for admission into the Math major.
 **'''1.''' What is the distribution of the number of students admitted?  Use SOCR to graph and print this distribution.  What is the shape of this distribution?  What is the mean and standard deviation of this distribution?
@@ Line 50: / Line 50: @@
 **'''4.''' Use the normal distribution to approximate the probability of part (2) (do not forget the continuity correction).  What is the error of the approximation?
-*'''Exercise 5:'''   The number of pine trees per acre in a forest follows the Poisson distribution with mean 30.
+*'''Exercise 6:'''   The number of pine trees per acre in a forest follows the Poisson distribution with mean 30.
 **'''1.''' Graph and print this distribution.

@@ Line 20: / Line 20: @@
 P(X \le x)=1-e^{-\lambda x}
 </math>
 *'''Exercise 2:'''   Use SOCR to graph and print the distribution of <math> X \sim N(20, 3) </math>.  Show on the graph the following points:  <math>\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma </math>.  How many standard deviations from the mean is the value <math> x=27.5 </math>?

SOCR EduMaterials Activities Continuous Distributions - Revision history

Rgidwani: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

Rgidwani: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

Rgidwani: changed the 30% in exercise 4 from a to a to keep uniform text.

IvoDinov at 00:33, 29 December 2006

Nchristo: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

Nchristo: /* This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson. */

Nchristo at 04:41, 25 November 2006

Rgidwani: changed the 30% in exercise 4 from a $to a to keep uniform text.$