Probstat/week6 practice 1 - ประวัติรุ่นแก้ไข

Jittat: /* Uniform distribution in 2D */

2014-09-23T01:30:27Z

Uniform distribution in 2D

Jittat: /* Experiments on continuous random variables */

2014-09-23T01:24:07Z

Experiments on continuous random variables

Jittat: /* Uniform distribution */

2014-09-23T01:14:39Z

Uniform distribution

Jittat: /* Generating random numbers under various distributions */

2014-09-23T01:10:28Z

Generating random numbers under various distributions

Jittat: /* Experiments on continuous random variables */

2014-09-23T01:10:12Z

Experiments on continuous random variables

Jittat: /* Experiments on continuous random variables */

2014-09-23T00:56:06Z

Experiments on continuous random variables

Jittat เมื่อ 18:21, 22 กันยายน 2557

2014-09-22T18:21:25Z

Jittat: /* Variances and distribution */

2014-09-22T18:19:06Z

Variances and distribution

Jittat: /* Variances and distribution */

2014-09-22T18:07:10Z

Variances and distribution

Jittat: หน้าที่ถูกสร้างด้วย ': ''This is part of probstat.'' In this practice, we will try a few experiments that shows distributions of random variables with ...'

2014-09-22T14:00:25Z

หน้าที่ถูกสร้างด้วย ': ''This is part of probstat.'' In this practice, we will try a few experiments that shows distributions of random variables with ...'

หน้าใหม่

: ''This is part of [[probstat]].''

In this practice, we will try a few experiments that shows distributions of random variables with varying variances.

== Variances and distribution ==
In each of the following experiments, you should find the expectation and the variance of the specified random variable. Then perform at least 10,000 experiments to generate the histogram of the distribution of the random variable.

=== Set 1: uniform distribution ===

=== Set 2: +1, -1 ===

=== Set 3: imbalance distribution ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:30, 23 กันยายน 2557
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	Try to write a program that generate these points and plot them on a 2D scatter chart.		Try to write a program that generate these points and plot them on a 2D scatter chart.

−	2.	+	2. We would like to generate a set of 1000 random points in 2D that are uniformly distributed in a circle of radius 1 centered at (0,0). (See image below.)
		+
		+	[[Image:Uniform-cir.png]]
		+
		+	How can we do that? Are you sure that the method that you suggest produces a uniform distribution?

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:24, 23 กันยายน 2557
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	=== Normal/Gaussian distribution ===		=== Normal/Gaussian distribution ===
		+	In this section, we will experiment with [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution]. Let random variable '''X''' represent a real number randomly distributed according to the normal distribution whose mean is 0 and standard deviation is 1.
		+
		+	1. Generate '''X''' for 10,000 times. What is the maximum and the minimum that you see?
		+
		+	2. Plot the histogram of '''X'''. (Generate '''X''' for 10,000 times, use about 100 buckets.)
		+
		+	3. Estimate '''E[X]'''.
		+
		+	4. Estimate '''Var(X)'''.

	=== Uniform distribution in 2D ===		=== Uniform distribution in 2D ===
		+	1. We want to generate a set of 1000 random points in 2D that are uniformly distributed in a 1x1 rectangle whose corners are (0,0), (0,1), (1,0), and (1,1). (An example is shown below.)
		+
		+	[[Image:Uniform-rec.png]]
		+
		+	Try to write a program that generate these points and plot them on a 2D scatter chart.
		+
		+	2.

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 === Uniform distribution ===
 === Normal/Gaussian distribution ===
 === Uniform distribution in 2D ===

@@ แถว 47: / แถว 47: @@
 '''Python:''' Python's [https://docs.python.org/2/library/random.html random] package provide many random number generators.
-* '''<tt>random.random()'''  - Return the next random floating point number in the range [0.0, 1.0).
+* '''<tt>random.random()</tt>'''  - Return the next random floating point number in the range [0.0, 1.0).
-* '''<tt>random.gauss(mu,sigma)''' - Gaussian distribution. mu is the mean, and sigma is the standard deviation.  In our experiment, we shall use mu = 0, sigma = 1.
+* '''<tt>random.gauss(mu,sigma)</tt>''' - Gaussian distribution. mu is the mean, and sigma is the standard deviation.  In our experiment, we shall use mu = 0, sigma = 1.
 '''Java:''' Use the Java [http://docs.oracle.com/javase/7/docs/api/java/util/Random.html Random] class.    You have to create an instance of the class, then call these methods:

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:10, 23 กันยายน 2557
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	In this practice, we will experiment on a few important continuous random variables (before we actually learn about it).		In this practice, we will experiment on a few important continuous random variables (before we actually learn about it).
		+
		+	=== Generating random numbers under various distributions ===
		+	In this practice we will generate uniform random variables and normally distributed random variables. The function/method for generating them in Python and Java is described below.
		+
		+	'''Python:''' Python's [https://docs.python.org/2/library/random.html random] package provide many random number generators.
		+
		+	* '''<tt>random.random()''' - Return the next random floating point number in the range [0.0, 1.0).
		+	* '''<tt>random.gauss(mu,sigma)''' - Gaussian distribution. mu is the mean, and sigma is the standard deviation. In our experiment, we shall use mu = 0, sigma = 1.
		+
		+	'''Java:''' Use the Java [http://docs.oracle.com/javase/7/docs/api/java/util/Random.html Random] class. You have to create an instance of the class, then call these methods:
		+
		+	* '''<tt>nextDouble()</tt>''' - Returns the next pseudorandom, uniformly distributed double value between 0.0 and 1.0 from this random number generator's sequence.
		+	* '''<tt>nextGaussian()</tt>''' - Returns the next pseudorandom, Gaussian ("normally") distributed double value with mean 0.0 and standard deviation 1.0 from this random number generator's sequence.
		+
		+	=== Uniform distribution ===
		+
		+	=== Normal/Gaussian distribution ===
		+
		+	=== Uniform distribution in 2D ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 00:56, 23 กันยายน 2557
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	== Experiments on continuous random variables ==		== Experiments on continuous random variables ==
		+	Don't be afraid of continuity!
		+
		+	In this practice, we will experiment on a few important continuous random variables (before we actually learn about it).

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 18:21, 22 กันยายน 2557
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	3.3 We pick two numbers 0 and 10 at random for 10 times (with replacement). For each trial, the probability that we pick 0 is 0.9 and the probability that we pick 10 is 0.1. Let '''Y''' be sum of the numbers that we pick. Let '''X = Y/10'''. Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.		3.3 We pick two numbers 0 and 10 at random for 10 times (with replacement). For each trial, the probability that we pick 0 is 0.9 and the probability that we pick 10 is 0.1. Let '''Y''' be sum of the numbers that we pick. Let '''X = Y/10'''. Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
		+
		+	== Experiments on continuous random variables ==

@@ แถว 11: / แถว 11: @@
 '''Hint:''' for question 1.2 - 1.5, finding '''Var(X)''' 'indirectly' using the result from 1.1 might be an easier approach.
-.2 We pick randomly an integer from the set {0,1,2,...,10} for 2 times.  Let '''Y''' be the sum of the 2 numbers that we picked.  Let '''X = Y/2'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
+.2 We pick randomly an integer from the set {0,1,2,...,10} for 2 times (with replacement).  Let '''Y''' be the sum of the 2 numbers that we picked.  Let '''X = Y/2'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
-.3 We pick randomly an integer from the set {0,1,2,...,10} for 3 times.  Let '''Y''' be the sum of the 3 numbers that we picked.  Let '''X = Y/3'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
+.3 We pick randomly an integer from the set {0,1,2,...,10} for 3 times (with replacement).  Let '''Y''' be the sum of the 3 numbers that we picked.  Let '''X = Y/3'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
-.4 We pick randomly an integer from the set {0,1,2,...,10} for 5 times.  Let '''Y''' be the sum of the 5 numbers that we picked.  Let '''X = Y/5'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
+.4 We pick randomly an integer from the set {0,1,2,...,10} for 5 times (with replacement).  Let '''Y''' be the sum of the 5 numbers that we picked.  Let '''X = Y/5'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
-.5 We pick randomly an integer from the set {0,1,2,...,10} for 10 times.  Let '''Y''' be the sum of the 10 numbers that we picked.  Let '''X = Y/10'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
+.5 We pick randomly an integer from the set {0,1,2,...,10} for 10 times (with replacement).  Let '''Y''' be the sum of the 10 numbers that we picked.  Let '''X = Y/10'''.  Find '''E[X]''', '''Var(X)''', and perform random experiments to plot '''X''' histogram.
 === Set 2: +1, -1 ===
 === Set 3: imbalance distribution ===