Statistics

Random variables are always generated with a particular pattern of probability attached to them. Thus, based on the pattern of probability for the different values of random variable, we can distinguish them. Once we know these probability distributions and their properties, and if any random variable fits in a probability distribution, so we will see what is probability mass function and probability density function A probability mass function (PMF)also called a frequency function  gives you probabilities for   discrete random variables. “Random variables” are variables from experiments like dice rolls, choosing a number out of a hat, or getting a high score on a test. The “discrete” part means that there’s a set number of outcomes. For example, you can only roll a 1, 2, 3, 4, 5, or 6 on a die. for example pmf for binomial distribution... Probability Mass Function Probability mass function is a function of defining a discrete probability dist...

  Rao-Blackwell Theorem  Calyampudi Radhakrishna Rao was born on 10 September 1920, in a  small town in south India called Huvvinna Hadagalli, then in the integrated  Madras Province of British India but now in the state of Karnataka.  C. R. Rao has received numerous  awards for his precious work in statistics, including the Guy Medal  in Silver of the Royal Statistical Society, London (1965), the Megnadh  Saha Medal of the Indian National Science Academy (1969), the Jagdish  Chandra Bose Gold Medal and cash award (1979), the Silver Plate bearing  the monogram of the Andhra Pradesh Academy of Sciences (1984), the S.  S. Wilks Medal of the American Statistical Association (1989), and the  Mahalanobis Birth Centenary Gold Medal awarded by the Indian Science  Congress (1996). The government of India honored C. R. Rao in 1968 by  awarding him the title of Padma Bhushan, a high civilian award. Let X and Y be ra...

  Laws of probability and conditional probability As in last post we have seen sample space and events in probability,today we will see some rules of probability  and conditional probability  To see laws of probability we will require some knowledge about independence and some terminologies regarding events. 1)Exhaustive events :The total number of possible outcomes of random experiment is known as the exhaustive events. Example:If we roll a die,there are 6 exhaustive cases since any number can appear uppermost 2)Favorable events :The outcome which make necessary the happening of an event in a trial are called favorable events. Example:let us consider two dies are rolled,then the number of favorable events getting sum 3 is 2,that is (1,2),(2,1) 3)Mutually exclusive events(Disjoint events) :The events are said to be mutually exclusive or disjoint if they cannot both occur at same time. Example:Outcome of single coin toss (either head or tail) 4)Equally likely events :...

  Sample space and Event Sample space:  A collection of all possible outcomes of an experiment is called sample space.It is denoted by  Ω. Example: A coin is tossed   Ω ={H,T} Where H=head, T=tail.   Event: The collection of all outcomes favorable to phenomenon is called an event. It is denoted by capital letters A,B,C.. Example: Event A is containing at least one head,if two coins are tossed.   Ω ={HH,HT,TH,TT} A={HH,HT,TH} (1) Elementary event: An event containing only one element is called as elementary or simple event. Example: Event A is getting both heads ,if two coins are tossed.   Ω ={HH,HT,TH,TT} A={HH} (2) Compound event: An event which contains more then one sample point of sample space is called a compound event. Example: Event A is containing both head and tail,if two coins are tossed. Ω ={HH,HT,TH,TT} A={HT,TH} (3)Sure event or certain event: An event containing all the points of sample space is called sure event. Example:  In an exp...

Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values. The above formula are given by prof.karl Pearson which are known as 'covexity of frequency curve' or simply kurtosis. These formula will give us idea about flatness or peakedness.It is measured by above  coefficients.If value of gamma2 is more  than zero then the curve is leptokurtic,if value equals to zero then it is mesokurtic or normal curve and if value of gamma 2 is less than zero than curve said to be platykurtic. 1)Mesokurtic Data that follows a mesokurtic distribution shows an excess kurtosis of zero or close to zero. This means that if the data follows a normal distribution, it follows a mesokurtic distribution. 2. Leptokurtic Leptokurtic indicates a positive excess kurtosis. The leptokurtic distribution shows h...

Mode : Mode is an important measure of central tendency.Ya Lun Chou statement about the mode is ,"Mode is that values of series which appears most frequently than any other." In simple  words,"The mode is a value which has highest frequency." The mode may not be unique .that is there are several values or classes with the same highest frequency. I) Mode for discrete data:      Mode of sample is the observation with highest frequency. Example: A sample is 4,8,3,8,3,9,3,7,11,3,12,11 Here number 3 has maximum frequency(4). Hence given sample has mode 3. II)Mode for grouped data:      Mode of grouped data is the class centre of the modal class,while the modal class is a class which has highest frequency. Example Here 10-20 class has maximum frequency (15). Hence 10-20 is modal class. Here L=10,f1=15,fo=5,f2=8,h=20-10= 10 Mode=10+[(15-5)/(30-5-8)]×10 Mode=10+(10/17)×10 Mode=10+5.88 Mode=15.88 For example,In clothing Store,they operate their business to incl...