Specifically, no more than 1\/k<\/em>2 of the distribution’s values can be more than k<\/em> standard deviations<\/a> away from the mean<\/p>\n equivalently, at least 1 \u2212 1\/k<\/em>2 of the distribution’s values are within k<\/em> standard deviations of the mean<\/p>\n In statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined.”<\/p>\n Ref: https:\/\/en.wikipedia.org\/wiki\/Chebyshev%27s_inequality<\/a><\/p>\n Let X<\/em> (integrable) be a random variable<\/a> with finite expected value<\/a> \u03bc<\/em> and finite non-zero variance<\/a> \u03c3<\/em>2. Then for any real number<\/a> k<\/em> > 0,<\/p>\n Only the case As an example, using “Markov’s inequality (and other similar inequalities) relate probabilities to expectations<\/a>, and provide (frequently loose but still useful) bounds for the cumulative distribution function<\/a> of a random variable.”<\/p>\n “If X is a nonnegative random variable and a<\/em> > 0, then the probability that X is at least a is at most the expectation of X divided by a:[1]<\/a><\/p>\n Let {\\displaystyle a={\\tilde {a}}\\cdot \\operatorname {E} (X)} “<\/p>\n Ref: https:\/\/en.wikipedia.org\/wiki\/Markov%27s_inequality<\/a><\/p>\n Check Null Hypothesis concept as well as Chi Square Test here: http:\/\/bangla.salearningschool.com\/recent-posts\/important-basic-concepts-statistics-for-big-data\/<\/a><\/p>\n Chi-Square Statistic:<\/strong><\/p>\n “A chi square<\/strong> (\u03c72) statistic<\/strong> is a test that measures how expectations compare to actual observed data (or model results).”<\/p>\n https:\/\/www.investopedia.com\/terms\/c\/chi-square-statistic.asp<\/a><\/p>\n “What does chi square test tell you?<\/p>\n The Chi<\/strong>–square test<\/strong> is intended to test<\/strong> how likely it is that an observed distribution is due to chance. It is also called a “goodness of fit” statistic, because it measures how well the observed distribution of data fits with the distribution that is expected if the variables are independent.”<\/p>\n https:\/\/www.ling.upenn.edu\/~clight\/chisquared.htm<\/a><\/p>\n “In probability theory<\/a> and statistics<\/a>, the chi-square distribution<\/strong> (also chi-squared<\/strong> or \u03c7<\/em>2-distribution<\/strong>) with k degrees of freedom<\/a> is the distribution of a sum of the squares of k independent<\/a> standard normal<\/a> random variables. The chi-square distribution is a special case of the gamma distribution<\/a> and is one of the most widely used probability distributions<\/a> in inferential statistics<\/a>, notably in hypothesis testing<\/a> and in construction of confidence intervals<\/a>.[2]<\/a>[3]<\/a>[4]<\/a>[5]<\/a> When it is being distinguished from the more general noncentral chi-square distribution<\/a>, this distribution is sometimes called the central chi-square distribution<\/strong>.”: https:\/\/en.wikipedia.org\/wiki\/Chi-squared_distribution<\/a><\/p>\n “A chi-squared test<\/strong>, also written as \u03c7<\/em>2 test<\/strong>, is any statistical hypothesis test<\/a> where the sampling distribution<\/a> of the test statistic is a chi-squared distribution<\/a> when the null hypothesis<\/a> is true. Without other qualification, ‘chi-squared test’ often is used as short for Pearson’s<\/em> chi-squared test<\/a>. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.”: https:\/\/en.wikipedia.org\/wiki\/Chi-squared_test<\/a><\/p>\n “<\/p>\n Learn<\/p>\n “<\/p>\n https:\/\/machinelearningmastery.com\/statistical-significance-tests-for-comparing-machine-learning-algorithms\/<\/a><\/p>\n Probability Axioms (I am not convinced that the following is the\u00a0 best way to say)<\/strong><\/p>\n “<\/strong><\/p>\n “<\/strong><\/p>\n https:\/\/plus.maths.org\/content\/maths-minute-axioms-probability<\/a><\/p>\n 1. Probability is non-negative<\/p>\n 2. P{S} = 1<\/p>\n 3. Probability is additive<\/p>\n If A and B are two mutually exclusive (independent) events<\/p>\n P (A U B) = P(A) + P(B)<\/p>\n P (A intersection B) = empty = 0 . [nothing common]<\/p>\n P{A} = 1 – P'(A)<\/p>\n P{phi = empty} = 0<\/p>\n What does probability density function mean?<\/strong><\/p>\n “Probability density function<\/strong> (PDF) is a statistical expression that defines a probability distribution<\/strong> for a continuous random variable as opposed to a discrete random variable. When the PDF is graphically portrayed, the area under the curve will indicate the interval in which the variable will fall” https:\/\/www.investopedia.com\/terms\/p\/pdf.asp<\/a><\/p>\n “A probability density function is most commonly associated with absolutely continuous<\/a> univariate distributions<\/a>. A random variable<\/a> Hence, if and Intuitively, one can think of Jump to navigation<\/a>Jump to search<\/a> “In probability<\/a> and statistics<\/a>, a probability mass function<\/strong> (PMF<\/strong>) is a function that gives the probability that a discrete random variable<\/a> is exactly equal to some value.[1]<\/a> Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a discrete probability distribution<\/a>, and such functions exist for either scalar<\/a> or multivariate random variables<\/a> whose domain<\/a> is discrete.<\/p>\n A probability mass function differs from a probability density function<\/a> (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated<\/a> over an interval to yield a probability.[2]<\/a><\/p>\n The value of the random variable having the largest probability mass is called the mode<\/a>.”https:\/\/en.wikipedia.org\/wiki\/Probability_mass_function<\/a><\/p>\n Here, we will discuss mixed<\/em> random variables. These are random variables that are neither discrete nor continuous, but are a mixture of both. In particular, a mixed random variable has a continuous part and a discrete part.<\/p>\n https:\/\/www.probabilitycourse.com\/chapter4\/4_3_1_mixed.php<\/a> . Also check the examples from here<\/p>\n Expected values of a random variable<\/strong> The \u201cmoments<\/strong>\u201d of a random variable<\/strong><\/p>\n The \u201cmoments<\/strong>\u201d of a random variable<\/strong> (or of its distribution) are expected values of powers or related functions of the random variable<\/strong>. The rth moment<\/strong> of X is E(Xr). In particular, the first moment<\/strong> is the mean, \u00b5X = E(X). The mean is a measure of the \u201ccenter\u201d or \u201clocation\u201d of a distribution<\/p>\n http:\/\/homepages.gac.edu\/~holte\/courses\/mcs341\/fall10\/documents\/sect3-3a.pdf<\/a><\/p>\n Joint distributions<\/p>\n “Joint distributions Notes: Below X and Y are assumed to be continuous random variables. This case is, by far, the most important case. Analogous formulas, with sums replacing integrals and p.m.f.\u2019s instead of p.d.f.\u2019s, hold for the case when X and Y are discrete r.v.\u2019s. Appropriate analogs also hold for mixed cases (e.g., X discrete, Y continuous), and for the more general case of n random variables X1, . . . , Xn.<\/p>\n \u2022 Joint cumulative distribution function (joint c.d.f.): F(x, y) = P(X \u2264 x, Y \u2264 y)”<\/p>\nProbabilistic statement[edit<\/a>]<\/span><\/h3>\n
<\/p>\n
is useful. When
the right-hand side
and the inequality is trivial as all probabilities are \u2264 1.<\/p>\n
shows that the probability that values lie outside the interval
does not exceed
.<\/p>\n
Ref: https:\/\/en.wikipedia.org\/wiki\/Chebyshev%27s_inequality<\/a><\/h3>\n
“Markov’s inequality<\/h3>\n
Statement<\/h2>\n
<\/p>\n
); then we can rewrite the previous inequality as<\/p>\n
Statistical Significance Tests for Comparing Machine Learning Algorithms<\/h1>\n
\n
\n
has density
, where
is a non-negative Lebesgue-integrable<\/a> function, if:
<\/p>\n
is the cumulative distribution function<\/a> of
, then:<\/p>\n
<\/p>\n
is continuous at
<\/p>\n
<\/p>\n
as being the probability of
falling within the infinitesimal interval<\/a>
.”
https:\/\/en.wikipedia.org\/wiki\/Probability_density_function<\/a><\/p>\nProbability mass function<\/h1>\n
![\"\"](\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/85\/Discrete_probability_distrib.svg\/220px-Discrete_probability_distrib.svg.png\")
<\/a>
The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.<\/p>\n4.3.1 Mixed Random Variables<\/h2>\n
The expected value of a discrete random variable<\/strong> is the probability-weighted average of all its possible values. In other words, each possible value the random variable<\/strong> can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value.
https:\/\/en.wikipedia.org\/wiki\/Expected_value<\/a><\/p>\n