C03 COMMON UNIVARIATE RANDOM VARIABLES

UNIFORM DISTRIBUTION  #

The continuous uniform distribution is defined over a range that spans between some lower limit, a, and some upper limit, b, which serve as the parameters of the distribution.  

Properties: 

For all a ≤ X1 < x2 ≤ b  

P(X < a or X > b) = 0  

P(x1 ≤ X ≤ x2) = (x2 — x1/(b — a).  

The probability density function for a continuous uniform distribution is expressed as : 

            f(x) = 1/(b-a) 

The mean and variance, respectively, of a uniform distribution are: 

            E(x) = (a+b)/2 

        Var(x) = (b-a)2/12 

BERNOULLI DISTRIBUTION  #

A Bernoulli distributed random variable only has two possible outcomes. The outcomes can be defined as either a “success” or a “failure.”  

Mean = μx = p 

Variance = p (1-p) 

 

THE BINOMIAL DISTRIBUTION  #

A binomial random variable may be defined as the number of “successes” in a given number of trials, whereby the outcome can be either “success” or “failure.”,The binomial probability function defines the probability of x successes in n trials. It can be expressed using the following formula: 

    P(x) = [n!/{(n-x)!x!}]px(1-p)n-x 

In TI BA II calculator first term is solved using nCr Function.  

 Expected value of X = E(X) = np 

 Variance of X = np(l — p) = npq 

POISSON DISTRIBUTION  #

The Poisson distribution is another discrete probability distribution. While the Poisson random variable X refers to the number of successes per unit, the parameter lambda (λ) refers to the average or expected number of successes per unit.  

        P(X=x) = (λx e-λ)/ x! 

Both mean & variance are equal to the parameter λ. 

NORMAL DISTRIBUTION  #

The normal distribution has the following key properties:  

• X is normally distributed with mean μ and variance σ2  
• Skewness = 0, meaning the normal distribution is symmetric about its mean,  

• Kurtosis = 3; this is a measure of how flat the distribution is.  
• A linear combination of normally distributed independent random variables is also normally distributed.  
• The probabilities of outcomes further above and below the mean get smaller and smaller but do not go to zero. 

Remember values and percentage probability covered for each SD give in above image for exam purpose.

STANDARD NORMAL DISTRIBUTION  #

A standard normal distribution (i.e., z-distribution) is a normal distribution that has been standardized so it has a mean of zero and a standard deviation of 1.  Standardization is the process of converting an observed value for a random variable to its z-value. 

LOGNORMAL DISTRIBUTION  #

The lognormal distribution is generated by the function ex, where x is normally distributed. The lognormal distribution is skewed to the right & is bounded from below by zero so that it is useful for modelling asset prices which never take negative values. 

STUDENTS T DISTRIBUTION   #

Student’s t-distribution is a bell-shaped probability distribution that is symmetrical about its mean. (Know how to get T value from table) 

Properties: 

• It is symmetrical.  
• It is defined by a single parameter, the degrees of freedom (df), where the degrees of freedom are equal to the number of sample observations minus 1, n — 1, for sample means.  
• It has more probability in the tails (fatter tails) than the normal distribution.  

• As the degrees of freedom (the sample size) gets larger, the shape of the t-distribution more closely approaches a standard normal distribution. 

CHI SQUARED DISTRIBUTION  #

The chi-square distribution is asymmetrical, bounded below by zero, and approaches the normal distribution in shape as the degrees of freedom increase. 

The chi-squared test statistic, χ, with n — 1 degrees of freedom, is computed as:   

 χ2n-1 =[(n-1)s2]/σ20 

Where: n= sample size, s2=sample variance,  

              σ20 = hypothesized value for the population variance 

F DISTRIBUTION  #

The hypotheses concerned with the equality of the variances of two populations are tested with an F-distributed test statistic. Hypothesis testing using a test statistic that follows an F-distribution is referred to as the F-test. The F-test is used under the assumption that the populations from which samples are drawn are normally distributed and that the samples are independent. The F– statistic is computed as:  

F=S21/S22F=S21/S22

 

Where: s21= variance of the sample of n1 observations drawn from Population 1  & s22  =  variance of the sample of n2 observations drawn from Population 2. 

For critical values from F distribution n1-1 and n2—1 degrees of freedom is used. 

Some additional properties of F distribution  

The F distribution approaches the normal distribution as the number of observations increases. 

A random variable t value squared with n-1 degrees of freedom is f distributed with 1 degree of freedom in the numerator and n-41 degrees of freedom in the denominator. 

There exists a relationship between the F and chi squard distribution such that  

F = X2 / # number of observations in numerator 

EXPONENTIAL DISTRIBUTION  #

Is used for waiting times, such as time take by the company to default.  

PDF  = f(x) = 1/beta X e-X/Beta , x>=0 

Beta is scale parameter, greater than zero and reciprocal of the rate parameter λ.  The rate at with company will default is example of rate parameter.  

Remember 

• Poisson distribution is used to evaluate number of default in a given period 
• Exponential distribution is used to calculate time take to default.  

Relationship of mean and variance can be established using Poisson.  

• Mean of Expo distribution is 1/λ 
• Variance of Expo Distribution is 1/λ2 

BETA DISTRIBUTION  #

Can be used for modelling default probabilities and recovery rate.  It is used in some credit models such as Creditriskmatrics.  

The mass of the beta distribution is located between the intervals zero and one. 

MIXTURE DISTRIBUTION  #

Mixture distribution is combination of one or more distribution and used to get unique PDF.  

It is used of data does not fit a predetermined distribution. It contains elements of both parametric and non parametric distributions.