Probability distribution

Smart Online tutoring classes : Smart Online tutoring classes Probability and statistics.

Types of distributions : Types of distributions Discrete distribution Continuous distribution

Discrete Distribution. : Discrete Distribution. Discrete uniform Bernoulli distribution Binomial distribution Geometric distribution Negative binomial (type I) Negative binomial (type II) Poisson distribution

Continuous Distribution : Continuous Distribution Continuous uniform Exponential Distribution Gamma distribution Chi-square distribution Beta distribution Normal (Gaussian) distribution Standard normal distribution Lognormal distribution Pareto distribution (with two parameters) Pareto distribution (with three parameters) Weibull distribution Burr distribution Last four distributions are not regular distributions.

Slide 5 : Discrete uniform Definition: Equal assignment (1/k) to all outcomes i.e. all outcomes are equally likely. Probability density function: P(X=x)= for all x=1, 2, …, k E(x) =µ= Variance (s^2)= µ and s^2 represent population mean and variance respectively. Standard deviation=Sqrt(variance)

Bernoulli Distribution : Bernoulli Distribution X=number of success Probability function P(X=x)=(p^x)*q^(1-x) x=0 , 1 and 0

Slide 7 : Binomial distribution Definition: Binomial distribution is the sum of n independent, identically Bernoulli trials With p probability of success. X=number of success in n trials Probability function: P(X=x)=nCx*p^x*q^(n-x) x=0, 1, 2,…,n ; 0

Geometric distribution : Geometric distribution Definition: This is the distribution where the variable of interest is to find out the Probability of first success in a sequence of independent identically Bernoulli trials. Like wise binomial there is no restriction on number of trials, there can be any number of trials. X=Number of trials on which the first success occurs. Probability function P(X=x)=p*q^(x-1) x=0, 1,… Mean µ= Variance (s^2)=

Negative binomial (Type 1) : Negative binomial (Type 1) Definition: The random variable X is the number of the trials on which kth success occurs, k is a positive integer. Probability distribution: P(X=x)=(x-1)C(k-1)*p^k*q(x-k) x=k, k+1, …. Mean µ= Variance (s^2)=

Negative binomial (type 2) : Negative binomial (type 2) Definition : The random variable Y represent number failures before kth success. Y=X-k , where X is defined in Negative binomial type 1 Probability distribution : P(Y=y)=(y+k-1)Cy*p^k*q^y, y=0, 1, … Mean µ= Variance (s^2)=

Poisson distribution : Poisson distribution Definition: This distribution provides the probability of number of events occurred in a Specific time interval, when the events occur one after another in time in a well defined manner. X=number of events. P(X=x)= x=0,1,… Mean µ= ? Variance (s^2)= ?

Continuous uniform : Continuous uniform Definition: A continuous random variable X will follows uniform distribution if its Defined between two numbers a and ß. Probability distribution of X is defined as f(x) = a

Exponential distribution : Exponential distribution Definition : A continuous random variable said to have follow exponential distribution If X has following probability distribution (PDF). PDF f(x)= , x>0 µ= s^2=

Gamma distribution : Gamma distribution Definition: The gamma family of distribution has 2 positive parameters with a and ?. The shape of the PDF depends on the values of these parameters and random variable vary from 0 to 8. PDF f(X)= , x>0 µ= s^2= Note :Finding probability for gamma through integration is a bit messy therefore we will use an approximation of gamma to chi square to find the probability for gamma distribution.

Chi-square distribution : Chi-square distribution Definition: Chi square distribution is a special case of gamma distribution with a=v/2 degree of freedom and ?=1/2. PDF f(x)= , x>0 Mean µ=v and s^2=2v Since integration of PDF of chi-square is not an easy task therefore to find the Probability for chi square we have extensive probability table.

Beta distribution : Beta distribution Definition : This is another family of distribution that have two parameters a and ß. The range of variable is 00 µ= s^2=

Normal distribution : Normal distribution Definition: This distribution has 2 parameters, which can conveniently be expressed directly as the mean µ and standard deviation s of the distribution. The distribution is Symmetrical about µ. PDF f(x)= for -8 < x < 8 Mean =µ, variance=s^2 and standard deviation= s Its PDF is also not easy to integrate, so again we will use table of standard normal Distribution to find probabilities.

Standard normal distribution : Standard normal distribution Standard normal distribution is a special case of normal distribution with mean 0 and standard deviation 1. PDF f(x)= f for -8 < x < 8 Mean µ=0 and standard deviation s=1

Lognormal distribution : Lognormal distribution Definition : A probability distribution in which the log of the random variable is normally distributed, meaning it conforms to a bell curve. Lognormal distribution are sometimes utilized in technical analysis. PDF f(x)= for 0