Probability distributions - Revision history

Trcs00 at 18:06, 1 June 2023

2023-06-01T18:06:26Z

Trcs00 at 18:05, 1 June 2023

2023-06-01T18:05:27Z

Trcs00: /* Sample space */

2023-06-01T18:01:32Z

‎Sample space

Trcs00: /* The Exponential Distribution */

2023-06-01T18:01:17Z

‎The Exponential Distribution

Trcs00 at 17:58, 1 June 2023

2023-06-01T17:58:05Z

Trcs00 at 17:54, 1 June 2023

2023-06-01T17:54:11Z

Trcs00 at 17:51, 1 June 2023

2023-06-01T17:51:45Z

Trcs00 at 17:49, 1 June 2023

2023-06-01T17:49:36Z

Trcs00 at 17:48, 1 June 2023

2023-06-01T17:48:53Z

Trcs00 at 17:47, 1 June 2023

2023-06-01T17:47:33Z

@@ Line 1: / Line 1: @@
-In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A.  by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
+In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
@@ Line 67: / Line 67: @@
 ==== Probability density function (PDF) ====
 To determine the distribution of a '''discrete''' random variable we can either provide its '''PMF''' or '''CDF'''. For '''continuous''' random variables, the CDF is well-defined so we can provide the '''CDF'''. However, the PMF does '''not''' work for continuous random variables, because for a continuous random variable ''P(X=x)=0'' for all x ∈ ℝ.
-*Instead, we can usually define the probability density function (PDF). The '''PDF''' is the density of probability rather than the probability mass. The concept is very similar to mass density in physics: its unit is probability per unit length.
+*Instead, we can usually define the probability density function (PDF). The '''PDF''' is the density of probability rather than the probability mass. The concept is very similar to mass density in physics: its unit is probability per unit length. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
-* The probability density function (PDF) is a function used to describe the probability distribution of a continuous random variable. Unlike discrete random variables, which have a countable set of possible values, continuous random variables can take on any value within a specified range.
+* The probability density function (PDF) is a function used to describe the probability distribution of a continuous random variable. Unlike discrete random variables, which have a countable set of possible values, continuous random variables can take on any value within a specified range. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
 * The PDF, denoted as ''f(x)'', represents the density of the probability distribution of a continuous random variable at a given point ''x''. It provides information about the likelihood of the random variable taking on a specific value or falling within a specific range of values.
-* Since the '''PDF is the derivative of the CDF''', the CDF can be obtained from PDF by integration
+* Since the '''PDF is the derivative of the CDF''', the CDF can be obtained from PDF by integration <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
 [[File:statszz.png|center|]]
 == Distribution Functions for Random Variables ==
-The distribution function provides important information about the probabilities associated with different values of a random variable. It can be used to calculate probabilities for specific events or to obtain other statistical properties of the random variable.
+The distribution function provides important information about the probabilities associated with different values of a random variable. It can be used to calculate probabilities for specific events or to obtain other statistical properties of the random variable. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
 *It gives the probability that the random variable takes on a value less than or equal to a given value.
-The distribution function of a random variable X, denoted as '''F(x)''', is defined as:
+The distribution function of a random variable X, denoted as '''F(x)''', is defined as: <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
 *F(x) = P(X ≤ x)
-where x is any real number, and P(X ≤ x) is the probability that the random variable X is less than or equal to x.
+where x is any real number, and P(X ≤ x) is the probability that the random variable X is less than or equal to x. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
 === Distribution Functions for Discrete Random Variables ===

@@ Line 12: / Line 12: @@
 === Random variable ===
-Random variable takes values from a sample space. In contrast, probabilities describe which values and set of values are more likely to be taken out of the sample space. Random variable must be quantified, therefore, it assigns a numerical value to each possible outcome in the sample space.
+Random variable takes values from a sample space. In contrast, probabilities describe which values and set of values are more likely to be taken out of the sample space. Random variable must be quantified, therefore, it assigns a numerical value to each possible outcome in the sample space. <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
-* For example, if the sample space for flipping a coin is {heads, tails}, then we can assign a random variable Y such that Y = 1 when heads land and Y = 0 when tails land. However, we can assign any number for these variables. 0 and 1 are just more convenient.
+* For example, if the sample space for flipping a coin is {heads, tails}, then we can assign a random variable Y such that Y = 1 when heads land and Y = 0 when tails land. However, we can assign any number for these variables. 0 and 1 are just more convenient.  <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 * Because random variables are defined to be functions of the outcome s, and because the outcome s is assumed to be random (i.e., to take on different values with different probabilities), it follows that the value of a random variable will itself be random (as the name implies).
@@ Line 21: / Line 21: @@
 * '''Exercise'''
-** Suppose that a coin is tossed twice so that the sample space is S = {HH, HT, TH, TT}. Let X represent the '''number of heads''' that can come up. With each sample point we can associate a number for X as shown in Table 1. Thus, for example, in the case of HH (i.e., 2 heads), X = 2 while for TH (1 head), X 􏰂= 1. It follows that X is a random variable.
+** Suppose that a coin is tossed twice so that the sample space is S = {HH, HT, TH, TT}. Let X represent the '''number of heads''' that can come up. With each sample point we can associate a number for X as shown in Table 1. Thus, for example, in the case of HH (i.e., 2 heads), X = 2 while for TH (1 head), X 􏰂= 1. It follows that X is a random variable. <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 [[File:Tablespace.png|center|[[Probability distributions#Sample Space - sample|Table 1. Sample Space]]]]
 === Expected value ===
-A very important concept in probability is that of the '''expected value''' of a random variable. For a discrete random variable ''X'' having the possible values ''x1, c, xn,'' the expectation of ''X'' is defined as:
+A very important concept in probability is that of the '''expected value''' of a random variable. For a discrete random variable ''X'' having the possible values ''x1, c, xn,'' the expectation of ''X'' is defined as: <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 [[File:ex.png|center|]]
@@ Line 216: / Line 216: @@
 The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. <ref name="probiz"> Pishro-Nik, H.  <i> Introduction to Probability, Statistics, and Random Processes</i>, published 2014, https://www.probabilitycourse.com/preface.php</ref>
-* It is defined by:
+* It is defined by: <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 [[File:exponentialol.png|center]]
-* An exponential distribution can often be used to model lifelengths. For example, a certain type of light bulb produced by a manufacturer might follow an ''Exponential(λ)'' distribution for an appropriate choice of ''λ''. The lifelength ''X'' of a randomly selected light bulb from those produced by this manufacturer has probability of lasting longer than ''x'' units of time can be calculated by following:
+* An exponential distribution can often be used to model lifelengths. For example, a certain type of light bulb produced by a manufacturer might follow an ''Exponential(λ)'' distribution for an appropriate choice of ''λ''. The lifelength ''X'' of a randomly selected light bulb from those produced by this manufacturer has probability of lasting longer than ''x'' units of time can be calculated by following:<ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 [[File:exponentialoll.png|center]]
-* The graph of the ''Exponential(λ)'' distribution depends on the ''λ'' paramater:
+* The graph of the ''Exponential(λ)'' distribution depends on the ''λ'' paramater: <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 [[File:exponentiagraph.png|center]]
@@ Line 234: / Line 234: @@
 [[File:gammafunnz.png|center]]
-* For the case α = 1 corresponds to the Exponential(λ) distribution: Gamma(1, λ) = Exponential(λ)
+* For the case α = 1 corresponds to the Exponential(λ) distribution: Gamma(1, λ) = Exponential(λ) <ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 ==== Example ====
-In the following graph, we can see the ''Exponential distribution'' function (solid line) and the ''Gamma function'' (dotted line) plotted:
+In the following graph, we can see the ''Exponential distribution'' function (solid line) and the ''Gamma function'' (dotted line) plotted:<ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 [[File:gammafunynz.png|center]]
 == References ==
 <references/>

@@ Line 7: / Line 7: @@
 == Terminology ==
 === Sample space  ===
-In probability theory, the sample space refers to the set of all possible outcomes of a random experiment. It is denoted by the symbol Ω (capital omega).
+In probability theory, the sample space refers to the set of all possible outcomes of a random experiment. It is denoted by the symbol Ω (capital omega).<ref name="probiiiz"> Casella, G., & Berger, R. L. <i> Statistical Inference. </i>, published 2021, Cengage Learning. </ref>
-* Let's consider an example of rolling a fair six-sided die. The sample space in this case would be {1, 2, 3, 4, 5, 6}, as these are the possible outcomes of the experiment. Each number represents the face of the die that may appear when it is rolled.
+* Let's consider an example of rolling a fair six-sided die. The sample space in this case would be {1, 2, 3, 4, 5, 6}, as these are the possible outcomes of the experiment. Each number represents the face of the die that may appear when it is rolled. <ref name="probiiiz"> Casella, G., & Berger, R. L. <i> Statistical Inference. </i>, published 2021, Cengage Learning. </ref>
 === Random variable ===

@@ Line 214: / Line 214: @@
 === The Exponential Distribution ===
 The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. <ref name="probiz"> Pishro-Nik, H.  <i> Introduction to Probability, Statistics, and Random Processes</i>, published 2014, https://www.probabilitycourse.com/preface.php</ref>
 * It is defined by:
 [[File:exponentialol.png|center]]

@@ Line 112: / Line 112: @@
 === The Uniform Distribution ===
-A continuous random variable ''X'' is said to have a '''Uniform''' distribution over the interval [a,b], shown as ''X ∼ Uniform(a,b)'', if its PDF is given by
+A continuous random variable ''X'' is said to have a '''Uniform''' distribution over the interval [a,b], shown as ''X ∼ Uniform(a,b)'', if its PDF is given by <ref name="probiz"> Pishro-Nik, H.  <i> Introduction to Probability, Statistics, and Random Processes</i>, published 2014, https://www.probabilitycourse.com/preface.php</ref>
 [[File:unifun.png|center]]
 The '''expected value''' is therefore
@@ Line 214: / Line 214: @@
 === The Exponential Distribution ===
-The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events.
+The exponential distribution is one of the widely used continuous distributions. It is often used to model the time elapsed between events. <ref name="probiz"> Pishro-Nik, H.  <i> Introduction to Probability, Statistics, and Random Processes</i>, published 2014, https://www.probabilitycourse.com/preface.php</ref>
 * It is defined by:
 [[File:exponentialol.png|center]]
@@ Line 225: / Line 225: @@
 ==== Example ====
-* Imagine you are at a store and are waiting for the next customer. In each millisecond, the probability that a new customer enters the store is very small. You can imagine that, in each millisecond, a coin (with a very small P(H) is tossed, and if it lands heads a new customers enters. If you toss a coin every millisecond, the time until a new customer arrives approximately follows an exponential distribution.
+* Imagine you are at a store and are waiting for the next customer. In each millisecond, the probability that a new customer enters the store is very small. You can imagine that, in each millisecond, a coin (with a very small P(H) is tossed, and if it lands heads a new customers enters. If you toss a coin every millisecond, the time until a new customer arrives approximately follows an exponential distribution.<ref name="probiz"> Pishro-Nik, H.  <i> Introduction to Probability, Statistics, and Random Processes</i>, published 2014, https://www.probabilitycourse.com/preface.php</ref>
 === The Gamma Distribution ===

@@ Line 1: / Line 1: @@
-In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A.  by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, vydáno 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
+In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A.  by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, published 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>
 ==Introduction==
-Probability is the science of uncertainty. It provides precise mathematical rules for understanding and analyzing our own ignorance. It does not tell us tomorrow’s weather or next week’s stock prices; rather, it gives us a framework for working with our limited knowledge and for making sensible decisions based on what we do and do not know.
+Probability is the science of uncertainty. It provides precise mathematical rules for understanding and analyzing our own ignorance. It does not tell us tomorrow’s weather or next week’s stock prices; rather, it gives us a framework for working with our limited knowledge and for making sensible decisions based on what we do and do not know.<ref name="probzz"> Evans, M. J., & Rosenthal, J. S.<i>Probability and Statistics: The Science of Uncertainty</i>, published 2009, WH Freeman. </ref>
 == Terminology ==

← Older revision		Revision as of 17:51, 1 June 2023
Line 1:		Line 1:
−	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">~~KŘÍŽ~~, ~~Oldřich~~, ~~Jiří NEUBAUER a Marek SEDLAČÍK~~. <i>~~Pravděpodobnost a náhodná veličina: učební text pro distanční studium. Brno~~: ~~Univerzita obrany~~</i>, vydáno ~~2008, 43s~~. ~~ISBN 978-80-7231-488-1~~</ref>	+	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">Spiegel, M. R., Schiller, J. T., & Srinivasan, A. by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. <i>Probability and Statistics : based on Schaum’s outline of Probability and Statistics</i>, vydáno 2001 https://ci.nii.ac.jp/ncid/BA77714681</ref>

← Older revision		Revision as of 17:49, 1 June 2023
Line 1:		Line 1:
−	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1.<ref name= </ref>	+	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. <ref name="probz">KŘÍŽ, Oldřich, Jiří NEUBAUER a Marek SEDLAČÍK. <i>Pravděpodobnost a náhodná veličina: učební text pro distanční studium. Brno: Univerzita obrany</i>, vydáno 2008, 43s. ISBN 978-80-7231-488-1</ref>
		+

	==Introduction==		==Introduction==

← Older revision		Revision as of 17:48, 1 June 2023
Line 1:		Line 1:
−	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1.<ref name="univobrany_43">KŘÍŽ, Oldřich, Jiří NEUBAUER a Marek SEDLAČÍK. <i>Pravděpodobnost a náhodná veličina: učební text pro distanční studium. Brno: Univerzita obrany</i>, vydáno 2008, 43s. ISBN 978-80-7231-488-1</ref>	+	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1.<ref name= </ref>

	==Introduction==		==Introduction==

← Older revision		Revision as of 17:47, 1 June 2023
Line 1:		Line 1:
−	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1.	+	In probability theory and statistics, a '''probability distribution''' is the mathematical function that gives the probabilities of occurrence of different possible '''outcomes''' for an experiment. In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1.<ref name="univobrany_43">KŘÍŽ, Oldřich, Jiří NEUBAUER a Marek SEDLAČÍK. <i>Pravděpodobnost a náhodná veličina: učební text pro distanční studium. Brno: Univerzita obrany</i>, vydáno 2008, 43s. ISBN 978-80-7231-488-1</ref>

	==Introduction==		==Introduction==