Category: Big Data & Analytics

Statistics – Discrete Series Arithmetic Mode ”; Previous Next When data is given along with their frequencies. Following is an example of discrete series − Items 5 10 20 30 40 50 60 70 Frequency 2 5 1 3 12 0 5 7 In discrete series, Arithmetic Mode can be determined by inspection and finding the variable which has the highest frequency associated with it. However, when there is very less difference between the maximum frequency and the frequency preceding it or succeeding it, then grouping table method is used. Example Problem Statement − Calculate Arithmetic Mode for the following discrete data − Items 14 36 45 70 105 145 Frequency 2 5 1 3 12 0 Solution − The Arithmetic Mode of the given numbers is 105 as the highest frequency,12 is associated with 105. Calculator Print Page Previous Next Advertisements ”;

Statistics – Permutation with Replacement ”; Previous Next Each of several possible ways in which a set or number of things can be ordered or arranged is called permutation Combination with replacement in probability is selecting an object from an unordered list multiple times. Permutation with replacement is defined and given by the following probability function: Formula ${^nP_r = n^r }$ Where − ${n}$ = number of items which can be selected. ${r}$ = number of items which are selected. ${^nP_r}$ = Ordered list of items or permutions Example Problem Statement: Electronic device usually require a personal code to operate. This particular device uses 4-digits code. Calculate how many codes are possible. Solution: Each code is represented by r=4 permutation with replacement of set of 10 digits{0,1,2,3,4,5,6,7,8,9} ${^{10}P_4 = (10)^4 \[7pt] = 10000 }$ Print Page Previous Next Advertisements ”;

Statistics – Gamma Distribution ”; Previous Next The gamma distribution represents continuous probability distributions of two-parameter family. Gamma distributions are devised with generally three kind of parameter combinations. A shape parameter $ k $ and a scale parameter $ theta $. A shape parameter $ alpha = k $ and an inverse scale parameter $ beta = frac{1}{ theta} $, called as rate parameter. A shape parameter $ k $ and a mean parameter $ mu = frac{k}{beta} $. Each parameter is a positive real numbers. The gamma distribution is the maximum entropy probability distribution driven by following criteria. Formula ${E[X] = k theta = frac{alpha}{beta} gt 0 and is fixed. \[7pt] E[ln(X)] = psi (k) + ln( theta) = psi( alpha) – ln( beta) and is fixed. }$ Where − ${X}$ = Random variable. ${psi}$ = digamma function. Characterization using shape $ alpha $ and rate $ beta $ Probability density function Probability density function of Gamma distribution is given as: Formula ${ f(x; alpha, beta) = frac{beta^alpha x^{alpha – 1 } e^{-x beta}}{Gamma(alpha)} where x ge 0 and alpha, beta gt 0 }$ Where − ${alpha}$ = location parameter. ${beta}$ = scale parameter. ${x}$ = random variable. Cumulative distribution function Cumulative distribution function of Gamma distribution is given as: Formula ${ F(x; alpha, beta) = int_0^x f(u; alpha, beta) du = frac{gamma(alpha, beta x)}{Gamma(alpha)}}$ Where − ${alpha}$ = location parameter. ${beta}$ = scale parameter. ${x}$ = random variable. ${gamma(alpha, beta x)} $ = lower incomplete gamma function. Characterization using shape $ k $ and scale $ theta $ Probability density function Probability density function of Gamma distribution is given as: Formula ${ f(x; k, theta) = frac{x^{k – 1 } e^{-frac{x}{theta}}}{theta^k Gamma(k)} where x gt 0 and k, theta gt 0 }$ Where − ${k}$ = shape parameter. ${theta}$ = scale parameter. ${x}$ = random variable. ${Gamma(k)}$ = gamma function evaluated at k. Cumulative distribution function Cumulative distribution function of Gamma distribution is given as: Formula ${ F(x; k, theta) = int_0^x f(u; k, theta) du = frac{gamma(k, frac{x}{theta})}{Gamma(k)}}$ Where − ${k}$ = shape parameter. ${theta}$ = scale parameter. ${x}$ = random variable. ${gamma(k, frac{x}{theta})} $ = lower incomplete gamma function. Print Page Previous Next Advertisements ”;

Statistics – Discrete Series Arithmetic Median ”; Previous Next When data is given along with their frequencies. Following is an example of discrete series − Items 5 10 20 30 40 50 60 70 Frequency 2 5 1 3 12 0 5 7 In case of a group having even number of distribution, Arithmetic Median is found out by taking out the Arithmetic Mean of two middle values after arranging the numbers in ascending order. Formula Median = Value of ($frac{N+1}{2})^{th} item$. Where − ${N}$ = Number of observations Example Problem Statement − Let”s calculate Arithmetic Median for the following discrete data − Items, ${X}$ 14 36 45 70 105 145 Frequency, ${f}$ 2 5 2 3 12 4 Comulative Frequency, ${C_f}$ 2 7 9 12 24 28 Terms 1-2 3-7 8-9 10-12 13-24 25-28 Solution − Based on the above mentioned formula, Arithmetic Median M will be − $M = Value of (frac{N+1}{2})^{th} item. \[7pt] , = Value of (frac{28+1}{2})^{th} item. \[7pt] , = Value of 14.5^{th} item. \[7pt] , = Value of (frac{14^{th} item + 15^{th} item}{2})\[7pt] , = (frac{105 + 105}{2}) , = {105}$ The Arithmetic Median of the given numbers is 2. In case of a group having even number of distribution, Arithmetic Median is the middle number after arranging the numbers in ascending order. Example Let”s calculate Arithmetic Median for the following discrete data − Items, ${X}$ 14 36 45 70 105 Frequency, ${f}$ 2 5 1 4 13 Comulative Frequency, ${C_f}$ 2 7 8 12 25 Terms 1-2 3-7 8-8 9-12 13-25 Given numbers are 25, an odd number thus middle number, 12th term is the Arithmetic Median. ∴ The Arithmetic Median of the given numbers is 70. Calculator Print Page Previous Next Advertisements ”;

Statistics – Probability Additive Theorem ”; Previous Next For Mutually Exclusive Events The additive theorem of probability states if A and B are two mutually exclusive events then the probability of either A or B is given by ${P(A or B) = P(A) + P(B) \[7pt] P (A cup B) = P(A) + P(B)}$ The theorem can he extended to three mutually exclusive events also as ${P(A cup B cup C) = P(A) + P(B) + P(C) }$ Example Problem Statement: A card is drawn from a pack of 52, what is the probability that it is a king or a queen? Solution: Let Event (A) = Draw of a card of king Event (B) Draw of a card of queen P (card draw is king or queen) = P (card is king) + P (card is queen) ${P (A cup B) = P(A) + P(B) \[7pt] = frac{4}{52} + frac{4}{52} \[7pt] = frac{1}{13} + frac{1}{13} \[7pt] = frac{2}{13}}$ For Non-Mutually Exclusive Events In case there is a possibility of both events to occur then the additive theorem is written as: ${P(A or B) = P(A) + P(B) – P(A and B)\[7pt] P (A cup B) = P(A) + P(B) – P(AB)}$ Example Problem Statement: A shooter is known to hit a target 3 out of 7 shots; whet another shooter is known to hit the target 2 out of 5 shots. Find the probability of the target being hit at all when both of them try. Solution: Probability of first shooter hitting the target P (A) = ${frac{3}{7}}$ Probability of second shooter hitting the target P (B) = ${frac{2}{5}}$ Event A and B are not mutually exclusive as both the shooters may hit target. Hence the additive rule applicable is ${P (A cup B) = P (A) + P(B) – P (A cap B) \[7pt] = frac{3}{7}+frac{2}{5}-(frac{3}{7} times frac{2}{5}) \[7pt] = frac{29}{35}-frac{6}{35} \[7pt] = frac{23}{35}}$ Print Page Previous Next Advertisements ”;

Statistics – Geometric Mean ”; Previous Next Geometric mean of n numbers is defined as the nth root of the product of n numbers. Formula ${GM = sqrt[n]{x_1 times x_2 times x_3 … x_n}}$ Where − ${n}$ = Total numbers. ${x_i}$ = numbers. Example Problem Statement: Determine the geometric mean of following set of numbers. 1 3 9 27 81 Solution: Step 1: Here n = 5 $ {GM = sqrt[n]{x_1 times x_2 times x_3 … x_n} \[7pt] , = sqrt[5]{1 times 3 times 9 times 27 times 81} \[7pt] , = sqrt[5]{3^3 times 3^3 times 3^4} \[7pt] , = sqrt[5]{3^{10}} \[7pt] , = sqrt[5]{{3^2}^5} \[7pt] , = sqrt[5]{9^5} \[7pt] , = 9 }$ Thus geometric mean of given numbers is $ 9 $. Print Page Previous Next Advertisements ”;

Statistics – Rayleigh Distribution ”; Previous Next The Rayleigh distribution is a distribution of continuous probability density function. It is named after the English Lord Rayleigh. This distribution is widely used for the following: Communications – to model multiple paths of densely scattered signals while reaching a receiver. Physical Sciences – to model wind speed, wave heights, sound or light radiation. Engineering – to check the lifetime of an object depending upon its age. Medical Imaging – to model noise variance in magnetic resonance imaging. The probability density function Rayleigh distribution is defined as: Formula ${ f(x; sigma) = frac{x}{sigma^2} e^{frac{-x^2}{2sigma^2}}, x ge 0 }$ Where − ${sigma}$ = scale parameter of the distribution. The comulative distribution function Rayleigh distribution is defined as: Formula ${ F(x; sigma) = 1 – e^{frac{-x^2}{2sigma^2}}, x in [0 infty}$ Where − ${sigma}$ = scale parameter of the distribution. Variance and Expected Value The expected value or the mean of a Rayleigh distribution is given by: ${ E[x] = sigma sqrt{frac{pi}{2}} }$ The variance of a Rayleigh distribution is given by: ${ Var[x] = sigma^2 frac{4-pi}{2} }$ Print Page Previous Next Advertisements ”;

Statistics – Exponential distribution ”; Previous Next Exponential distribution or negative exponential distribution represents a probability distribution to describe the time between events in a Poisson process. In Poisson process events occur continuously and independently at a constant average rate. Exponential distribution is a particular case of the gamma distribution. Probability density function Probability density function of Exponential distribution is given as: Formula ${ f(x; lambda ) = } $ $ begin {cases} lambda e^{-lambda x}, & text{if $x ge 0 $} \[7pt] 0, & text{if $x lt 0 $} end{cases} $ Where − ${lambda}$ = rate parameter. ${x}$ = random variable. Cumulative distribution function Cumulative distribution function of Exponential distribution is given as: Formula ${ F(x; lambda) = }$ $ begin {cases} 1- e^{-lambda x}, & text{if $x ge 0 $} \[7pt] 0, & text{if $x lt 0 $} end{cases} $ Where − ${lambda}$ = rate parameter. ${x}$ = random variable. Print Page Previous Next Advertisements ”;

Statistics – Gumbel Distribution ”; Previous Next Gumbel Distribution represents the distribution of extreme values either maximum or minimum of samples used in various distributions. It is used to model distribution of peak levels. For example, to show the distribution of peak temperatures of the year if there is a list of maximum temperatures of 10 years. Probability density function Probability density function of Gumbel distribution is given as: Formula ${ P(x) = frac{1}{beta} e^{[frac{x – alpha}{beta} – e^{frac{x – alpha}{beta}}]} }$ Where − ${alpha}$ = location parameter. ${beta}$ = scale parameter. ${x}$ = random variable. Cumulative distribution function Cumulative distribution function of Gumbel distribution is given as: Formula ${ D(x) = 1 – e^{-e^{frac{x – alpha}{beta}}}}$ Where − ${alpha}$ = location parameter. ${beta}$ = scale parameter. ${x}$ = random variable. Print Page Previous Next Advertisements ”;

Statistics – F distribution ”; Previous Next The F distribution (Snedecor”s F distribution or the Fisher–Snedecor distribution) represents continuous probability distribution which occurs frequently as null distribution of test statistics. It happens mostly during analysis of variance or F-test. Probability density function Probability density function of F distribution is given as: Formula ${ f(x; d_1, d_2) = frac{sqrt{frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1x+d_2)^{d_1+d_2}}}}{x beta (frac{d_1}{2}, frac{d_2}{2})} }$ Where − ${d_1}$ = positive parameter. ${d_2}$ = positive parameter. ${x}$ = random variable. Cumulative distribution function Cumulative distribution function of F distribution is given as: Formula ${ F(x; d_1, d_2) = I_{frac{d_1x}{d_1x+d_2}}(frac{d_1}{2}, frac{d_2}{2})}$ Where − ${d_1}$ = positive parameter. ${d_2}$ = positive parameter. ${x}$ = random variable. ${I} $ = lower incomplete beta function. Print Page Previous Next Advertisements ”;