Probstat/notes/random variables - ประวัติรุ่นแก้ไข

Jittat: /* Expectations */

2014-09-18T04:01:53Z

Expectations

Jittat เมื่อ 03:59, 18 กันยายน 2557

2014-09-18T03:59:03Z

Jittat: /* Expectations */

2014-09-18T03:53:50Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:53:30Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:18:33Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:17:29Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:16:15Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:12:51Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:12:16Z

Expectations

Jittat: /* Expectations */

2014-09-18T03:11:46Z

Expectations

@@ แถว 98: / แถว 98: @@
 From this definition, we can compute expectations related to random variables ''X'' and ''Y'' defined previously.
-'''EX1:''' <math>\mathrm{E}[X] = \left( 2\cdot\frac{1}{36} +
+{{กล่องเทา|'''EX1:''' <math>\mathrm{E}[X] = \left( 2\cdot\frac{1}{36} +
 \cdot\frac{2}{36} +
 \cdot\frac{3}{36} +
@@ แถว 109: / แถว 109: @@
 \cdot\frac{2}{36} +
 \cdot\frac{1}{36} \right) = 7.</math>
-'''EX2:''' <math>\mathrm{E}[Y] = 2\cdot\frac{1}{6} + 3\cdot\frac{1}{3} + 4\cdot\frac{1}{2} = \frac{10}{3}\approx 3.3333</math>.
+{{กล่องเทา|'''EX2:''' <math>\mathrm{E}[Y] = 2\cdot\frac{1}{6} + 3\cdot\frac{1}{3} + 4\cdot\frac{1}{2} = \frac{10}{3}\approx 3.3333</math>.}}
-'''EX3:''' We can also compute the expectation of some function of a random variable.  Suppose that we want to compute E[''Y''<sup>2</sup>].  We can look at all possible values of ''Y'' and take the average over the values of ''Y''<sup>2</sup> weighted by their probability.  I.e.,
+{{กล่องเทา|'''EX3:''' We can also compute the expectation of some function of a random variable.  Suppose that we want to compute E[''Y''<sup>2</sup>].  We can look at all possible values of ''Y'' and take the average over the values of ''Y''<sup>2</sup> weighted by their probability.  I.e.,
 <math>\mathrm{E}[Y^2] = 2^2\cdot P\{Y^2 = 2^2\} + 3^2\cdot P\{Y^2 = 3^2\} + 4^2\cdot P\{Y^2 = 4^2\}</math>
@@ แถว 119: / แถว 120: @@
 <math> = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}.</math>
-'''EX4:''' As another example, consider a random variable ''Z'' which is -1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3.  We can find its expectation:
+{{กล่องเทา|'''EX4:''' As another example, consider a random variable ''Z'' which is -1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3.  We can find its expectation:
 <math>\mathrm{E}[Z] = -1\cdot P\{Z=-1\} + 0\cdot P\{Z=0\} + 1\cdot P\{Z=1\}
@@ แถว 129: / แถว 131: @@
 <math>\mathrm{E}[Z^2] = (-1)^2\cdot P\{Z=-1\} + (0)^2\cdot P\{Z=0\} + (1)^2\cdot P\{Z=1\}
   = (-1)^2\cdot\frac{1}{3} + (0)^2\cdot\frac{1}{3} + (1)^2\cdot\frac{1}{3} = 2/3.</math>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:59, 18 กันยายน 2557
แถว 1:		แถว 1:
−	: ''This is part of [[probstat]].''	+	: ''This is part of [[probstat]]. These notes are meant to be used in complement to the video lectures. They only contain summary of the materials discussed in the video. Don't use them to avoid watching the clips please.''

	In many cases, after we perform a random experiment, we are interested in certain quantity from the outcome, not the actual outcome. In that case, we can define a '''random variable''', which is a function from the sample space to real numbers, to represent the random quantity that we are interested in.		In many cases, after we perform a random experiment, we are interested in certain quantity from the outcome, not the actual outcome. In that case, we can define a '''random variable''', which is a function from the sample space to real numbers, to represent the random quantity that we are interested in.

@@ แถว 91: / แถว 91: @@
 {{กล่องฟ้า|For a integer random variables ''X'', the '''expected value''' of ''X'' (or the '''expectation''' of ''X''), denoted by E[''X''] is defined as
-<math>\mathrm{E}[X] = \sum_{i=-\infty}^{\infty} i\cdot P\{X=i\}</math>.
+<center><math>\mathrm{E}[X] = \sum_{i=-\infty}^{\infty} i\cdot P\{X=i\}</math>.</center>
 }}

@@ แถว 89: / แถว 89: @@
 The whole point of having probability models is that we want to say "something" about the experiments without having to perform them or exhaustively trying all their possible outcomes.  For a given random variable, we would like to have "some number" that represents it on average.  From this motivation, we have the definition of the expectation as follows.
-For a integer random variables ''X'', the '''expected value''' of ''X'' (or the '''expectation''' of ''X''), denoted by E[''X''] is defined as
+{{กล่องฟ้า|For a integer random variables ''X'', the '''expected value''' of ''X'' (or the '''expectation''' of ''X''), denoted by E[''X''] is defined as
 <math>\mathrm{E}[X] = \sum_{i=-\infty}^{\infty} i\cdot P\{X=i\}</math>.
 This is just the weighted average of the possible values of ''X''.  (Note that each value ''i'' is weighted by its weight ''P{ X = i }''.)

@@ แถว 97: / แถว 97: @@
 From this definition, we can compute expectations related to random variables ''X'' and ''Y'' defined previously.
-. <math>\mathrm{E}[X] = \left( 2\cdot\frac{1}{36} +
+'''EX1:''' <math>\mathrm{E}[X] = \left( 2\cdot\frac{1}{36} +
 \cdot\frac{2}{36} +
 \cdot\frac{3}{36} +
@@ แถว 109: / แถว 109: @@
 \cdot\frac{1}{36} \right) = 7.</math>
-. <math>\mathrm{E}[Y] = 2\cdot\frac{1}{6} + 3\cdot\frac{1}{3} + 4\cdot\frac{1}{2} = \frac{10}{3}\approx 3.3333</math>.
+'''EX2:''' <math>\mathrm{E}[Y] = 2\cdot\frac{1}{6} + 3\cdot\frac{1}{3} + 4\cdot\frac{1}{2} = \frac{10}{3}\approx 3.3333</math>.
-. We can also compute the expectation of some function of a random variable.  Suppose that we want to compute E[''Y''<sup>2</sup>].  We can look at all possible values of ''Y'' and take the average over the values of ''Y''<sup>2</sup> weighted by their probability.  I.e.,
+'''EX3:''' We can also compute the expectation of some function of a random variable.  Suppose that we want to compute E[''Y''<sup>2</sup>].  We can look at all possible values of ''Y'' and take the average over the values of ''Y''<sup>2</sup> weighted by their probability.  I.e.,
 <math>\mathrm{E}[Y^2] = 2^2\cdot P\{Y^2 = 2^2\} + 3^2\cdot P\{Y^2 = 3^2\} + 4^2\cdot P\{Y^2 = 4^2\}</math>
@@ แถว 119: / แถว 119: @@
 <math> = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}.</math>
-. As another example, consider a random variable ''Z'' which is -1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3.  We can find its expectation:
+'''EX4:''' As another example, consider a random variable ''Z'' which is -1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3.  We can find its expectation:
 <math>\mathrm{E}[Z] = -1\cdot P\{Z=-1\} + 0\cdot P\{Z=0\} + 1\cdot P\{Z=1\}

@@ แถว 121: / แถว 121: @@
 . As another example, consider a random variable ''Z'' which is -1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3.  We can find its expectation:
-<math>\mathrm{E}[Z] = -1\cdot\frac{1}{3} + 0\cdot\frac{1}{3} + 1\cdot\frac{1}{3} = 0.</math>
+<math>\mathrm{E}[Z] = -1\cdot P\{Z=-1\} + 0\cdot P\{Z=0\} + 1\cdot P\{Z=1\}
 We can also find <math>\mathrm{E}[Z^2]</math>:
-<math>\mathrm{E}[Z^2] = (-1)^2\cdot\frac{1}{3} + (0)^2\cdot\frac{1}{3} + (1)^2\cdot\frac{1}{3} = 2/3.</math>
+<math>\mathrm{E}[Z^2] = (-1)^2\cdot P\{Z=-1\} + (0)^2\cdot P\{Z=0\} + (1)^2\cdot P\{Z=1\}

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:16, 18 กันยายน 2557
แถว 118:		แถว 118:

	<math> = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}.</math>		<math> = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}.</math>
		+
		+	4. As another example, consider a random variable ''Z'' which is -1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3. We can find its expectation:
		+
		+	<math>\mathrm{E}[Z] = -1\cdot\frac{1}{3} + 0\cdot\frac{1}{3} + 1\cdot\frac{1}{3} = 0.</math>
		+
		+	We can also find <math>\mathrm{E}[Z^2]</math>:
		+
		+	<math>\mathrm{E}[Z^2] = (-1)^2\cdot\frac{1}{3} + (0)^2\cdot\frac{1}{3} + (1)^2\cdot\frac{1}{3} = 2/3.</math>

@@ แถว 113: / แถว 113: @@
 . We can also compute the expectation of some function of a random variable.  Suppose that we want to compute E[''Y''<sup>2</sup>].  We can look at all possible values of ''Y'' and take the average over the values of ''Y''<sup>2</sup> weighted by their probability.  I.e.,
-<math>\mathrm{E}[Y^2] = 2^2\cdot P\{Y^2 = 2^2} + 3^2\cdot P\{Y^2 = 3^2} + 4^2\cdot P\{Y^2 = 4^2}
+<math>\mathrm{E}[Y^2] = 2^2\cdot P\{Y^2 = 2^2\} + 3^2\cdot P\{Y^2 = 3^2\} + 4^2\cdot P\{Y^2 = 4^2\}
-= 2^2\cdot P\{Y = 2} + 3^2\cdot P\{Y = 3} + 4^2\cdot P\{Y = 4} = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}
+= 2^2\cdot P\{Y = 2\} + 3^2\cdot P\{Y = 3\} + 4^2\cdot P\{Y = 4\} = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}
 </math>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:11, 18 กันยายน 2557
แถว 111:		แถว 111:
	2. <math>\mathrm{E}[Y] = 2\cdot\frac{1}{6} + 3\cdot\frac{1}{3} + 4\cdot\frac{1}{2} = \frac{10}{3}\approx 3.3333</math>.		2. <math>\mathrm{E}[Y] = 2\cdot\frac{1}{6} + 3\cdot\frac{1}{3} + 4\cdot\frac{1}{2} = \frac{10}{3}\approx 3.3333</math>.

−	3. We can also compute the expectation of some function of a random variable. Suppose that we want to compute E[''Y''<sup>2</sup>].	+	3. We can also compute the expectation of some function of a random variable. Suppose that we want to compute E[''Y''<sup>2</sup>]. We can look at all possible values of ''Y'' and take the average over the values of ''Y''<sup>2</sup> weighted by their probability. I.e.,
		+
		+	<math>\mathrm{E}[Y^2] = 2^2\cdot P\{Y^2 = 2^2} + 3^2\cdot P\{Y^2 = 3^2} + 4^2\cdot P\{Y^2 = 4^2}
		+	= 2^2\cdot P\{Y = 2} + 3^2\cdot P\{Y = 3} + 4^2\cdot P\{Y = 4} = 4\cdot\frac{1}{6} + 9\cdot\frac{1}{3} + 16\cdot\frac{1}{2} = 11\frac{2}{3}
		+	</math>