Expected Value - Revision history

Khanh at 18:10, 24 October 2021

2021-10-24T18:10:13Z

Lila at 19:30, 15 October 2021

2021-10-15T19:30:18Z

Lila at 19:27, 15 October 2021

2021-10-15T19:27:33Z

Lila at 19:27, 15 October 2021

2021-10-15T19:27:05Z

Lila: /* Examples */

2021-10-15T19:26:01Z

Examples

Lila at 19:25, 15 October 2021

2021-10-15T19:25:31Z

Johnraymond.yanez: Added Links to materials

2020-08-17T14:45:53Z

Added Links to materials

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Expected Value], Book Chapter
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Guided Notes]

← Older revision		Revision as of 19:30, 15 October 2021
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	:That is, the bet of $1 stands to lose <math>-\$\frac{1}{19}</math>, so its expected value is <math>-\$\frac{1}{19}.</math>		:That is, the bet of $1 stands to lose <math>-\$\frac{1}{19}</math>, so its expected value is <math>-\$\frac{1}{19}.</math>
		+
		+

	==Resources==		==Resources==

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-Let <math>X</math> be a (discrete) random variable with a finite number of finite outcomes <math>x_1, x_2, \ldots, x_k</math> occurring with probabilities <math>p_1, p_2, \ldots, p_k,</math> respectively. The '''expectation''' of <math>X</math> is defined as <ref name=":0" />
+Let <math>X</math> be a (discrete) random variable with a finite number of finite outcomes <math>x_1, x_2, \ldots, x_k</math> occurring with probabilities <math>p_1, p_2, \ldots, p_k,</math> respectively. The '''expectation''' of <math>X</math> is defined as
 :<math>\operatorname{E}[X] =\sum_{i=1}^k x_i\,p_i=x_1p_1 + x_2p_2 + \cdots + x_kp_k.</math>
@@ Line 7: / Line 7: @@
 If all outcomes <math>x_i</math> are equiprobable (that is, <math>p_1 = p_2 = \cdots = p_k</math>), then the weighted average turns into the simple average. On the other hand, if the outcomes <math>x_i</math> are not equiprobable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than others.
-[[File:Largenumbers.svg|An illustration of the convergence of sequence averages of rolls of a {{dice}} to the expected value of 3.5 as the number of rolls (trials) grows.|thumb|right|400 px]]
+[[File:Largenumbers.svg|An illustration of the convergence of sequence averages of rolls of a dice to the expected value of 3.5 as the number of rolls (trials) grows.|thumb|right|400 px]]
 ====Examples====

@@ Line 10: / Line 10: @@
 ====Examples====
-*Let <math>X</math> represent the outcome of a roll of a fair six-sided dice. More specifically, <math>X</math> will be the number of pips showing on the top face of the dice after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of {{frac2|1|6}}. The expectation of <math>X</math> is
+*Let <math>X</math> represent the outcome of a roll of a fair six-sided dice. More specifically, <math>X</math> will be the number of pips showing on the top face of the dice after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of 1/6. The expectation of <math>X</math> is
 :: <math>\operatorname{E}[X] = 1\cdot\frac16 + 2\cdot\frac16 + 3\cdot\frac16 + 4\cdot\frac16 + 5\cdot\frac16 + 6\cdot\frac16 = 3.5.</math>
 :If one rolls the dice <math>n</math> times and computes the average (arithmetic mean) of the results, then as <math>n</math> grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.
-*The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable <math>X</math> represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability {{frac2|1|38}} in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
+*The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable <math>X</math> represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability 1/38 in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
 :: <math>   \operatorname{E}[\,\text{gain from }\$1\text{ bet}\,] = -\$1 \cdot \frac{37}{38} + \$35 \cdot \frac{1}{38} = -\$\frac{1}{19}.</math>

@@ Line 10: / Line 10: @@
 ====Examples====
-*Let <math>X</math> represent the outcome of a roll of a fair six-sided {{dice}}. More specifically, <math>X</math> will be the number of pips showing on the top face of the {{dice}} after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of {{frac2|1|6}}. The expectation of <math>X</math> is
+*Let <math>X</math> represent the outcome of a roll of a fair six-sided dice. More specifically, <math>X</math> will be the number of pips showing on the top face of the dice after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of {{frac2|1|6}}. The expectation of <math>X</math> is
 :: <math>\operatorname{E}[X] = 1\cdot\frac16 + 2\cdot\frac16 + 3\cdot\frac16 + 4\cdot\frac16 + 5\cdot\frac16 + 6\cdot\frac16 = 3.5.</math>
-:If one rolls the {{dice}} <math>n</math> times and computes the average (arithmetic mean) of the results, then as <math>n</math> grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.
+:If one rolls the dice <math>n</math> times and computes the average (arithmetic mean) of the results, then as <math>n</math> grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.
 *The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable <math>X</math> represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability {{frac2|1|38}} in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
@@ Line 19: / Line 19: @@
 :That is, the bet of $1 stands to lose <math>-\$\frac{1}{19}</math>, so its expected value is <math>-\$\frac{1}{19}.</math>
 ==Resources==

← Older revision		Revision as of 19:25, 15 October 2021
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		+	Let <math>X</math> be a (discrete) random variable with a finite number of finite outcomes <math>x_1, x_2, \ldots, x_k</math> occurring with probabilities <math>p_1, p_2, \ldots, p_k,</math> respectively. The '''expectation''' of <math>X</math> is defined as <ref name=":0" />
		+
		+	:<math>\operatorname{E}[X] =\sum_{i=1}^k x_i\,p_i=x_1p_1 + x_2p_2 + \cdots + x_kp_k.</math>
		+
		+	Since <math>p_1 + p_2 + \cdots + p_k = 1,</math> the expected value is the weighted sum of the <math>x_i</math> values, with the probabilities <math>p_i</math> as the weights.
		+
		+	If all outcomes <math>x_i</math> are equiprobable (that is, <math>p_1 = p_2 = \cdots = p_k</math>), then the weighted average turns into the simple average. On the other hand, if the outcomes <math>x_i</math> are not equiprobable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than others.
		+
		+	[[File:Largenumbers.svg\|An illustration of the convergence of sequence averages of rolls of a {{dice}} to the expected value of 3.5 as the number of rolls (trials) grows.\|thumb\|right\|400 px]]
		+
		+	====Examples====
		+	*Let <math>X</math> represent the outcome of a roll of a fair six-sided {{dice}}. More specifically, <math>X</math> will be the number of pips showing on the top face of the {{dice}} after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of {{frac2\|1\|6}}. The expectation of <math>X</math> is
		+	:: <math>\operatorname{E}[X] = 1\cdot\frac16 + 2\cdot\frac16 + 3\cdot\frac16 + 4\cdot\frac16 + 5\cdot\frac16 + 6\cdot\frac16 = 3.5.</math>
		+
		+	:If one rolls the {{dice}} <math>n</math> times and computes the average (arithmetic mean) of the results, then as <math>n</math> grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.
		+
		+	*The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable <math>X</math> represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability {{frac2\|1\|38}} in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
		+	:: <math> \operatorname{E}[\,\text{gain from }\$1\text{ bet}\,] = -\$1 \cdot \frac{37}{38} + \$35 \cdot \frac{1}{38} = -\$\frac{1}{19}.</math>
		+
		+	:That is, the bet of $1 stands to lose <math>-\$\frac{1}{19}</math>, so its expected value is <math>-\$\frac{1}{19}.</math>
		+
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Expected_value Expected Value], Wikipedia
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Expected Value], Book Chapter		* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Expected Value], Book Chapter
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Guided Notes]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Guided Notes]

@@ Line 23: / Line 23: @@
 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Expected Value], Book Chapter
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Expected_Value/MAT1053_M12.2Expected_Value.pdf Guided Notes]