Some additional discrete random variable and its parameters
The discrete random variable with its probability mass function combines the distribution of the probability and depending on the nature of the discrete random variable the probability distribution may have different names like binomial distribution, Poisson distribution etc., as already we has seen the types of discrete random variable, binomial random variable and Poisson random variable with the statistical parameters for these random variables. Most of the random variables are characterized depending on the nature of probability mass function, now we will see some more type of discrete random variables and its statistical parameters.
Topics on Probability
Geometric Random variable and its distribution
A geometric random variable is the random variable which is assigned for the independent trials performed till the occurrence of success after continuous failure i.e if we perform an experiment n times and getting initially all failures n-1 times and then at the last we get success. The probability mass function for such a discrete random variable will be
In this random variable the necessary condition for the outcome of the independent trial is the initial all the result must be failure before success.
Thus in brief the random variable which follows above probability mass function is known as geometric random variable.
It is easily observed that the sum of such probabilities will be 1 as the case for the probability.
Thus the geometric random variable with such probability mass function is geometric distribution.
Expectation of Geometric random variable
As expectation is one of the important parameter for the random variable so the expectation for the geometric random variable will be
where p is the probability of success.
let the probability of failure be q=1-p
thus we get
Thus the expected value or mean of the given information we can follow by just inverse value of probability of success in geometric random variable.
Variance and standard deviation of the geometric random variable
In similar way we can obtain the other important statistical parameter variance and standard deviation for the geometric random variable and it would be
To obtain these values we use the relation
So let us calculate first
So we now write
thus we have
Negative Binomial Random Variable
This random falls in another discrete random variable because of the nature of its probability mass function, in the negative binomial random variable and in its distribution from n trial of an independent experiment r successes must be obtained initially
In other words a random variable with above probability mass function is negative binomial random variable with parameters (r,p), note that if we restrict r=1 the negative binomial distribution turns to geometric distribution, we can specifically check
Expectation, Variance and standard deviation of the negative binomial random variable
The expectation and variance for the negative binomial random variable will be
with the help of probability mass function of negative binomial random variable and definition of expectation we can write
here Y is nothing but the negative binomial random variable now put k=1 we will get
Thus for variance
Exxample: If a die is throw to get 5 on the face of die till we get 4 times this value find the expectation and variance.Sine the random variable associated with this independent experiment is negative binomial random variable for r=4 and probability of success p=1/6 to get 5 in one throw
as we know for negative binomial random variable
Hypergeometric random variable
If we particularly choosing a sample of size n from a total N having m and N-m two types then the random variable for first was selected have the probability mass function as
for example suppose we have a sack from which a sample of size n books taken randomly without replacement containing N books of which m are mathematics and N-m are physics, If we assign the random variable to denote the number of mathematics books selected then the probability mass function for such selection will be as per above probability mass function.
In other words the random variable with the above probability mass function is known to be the hypergeometric random variable.
Example: From a lot of some electronic components if 30% of the lots have four defective components and 70% have one defective, provided size of lot is 10 and to accept the lot three random components will be chosen and checked if all are non-defective then lot will be selected. Calculate that from the total lot what percent of lot get rejected.
here consider A is the event to accept the lot
N=10, m=4, n=3
for N=10, m=1, n=3
Thus the 46% lot will be rejected.
Expectation, Variance and standard deviation of the hypergeometric random variable
The expectation, variance and standard deviation for the hypergeometric random variable with parameters n,m, and N would be
or for the large value of N
and standard deviation is the square root of the variance.
By considering the definition of probability mass function of hypergeormetric function and the expectation we can write it as
here by using the relations and identities of the combinations we have
so it would be
here Y plays the role of hypergeometric random variable with respective parameters now if we put k=1 we will get
and for k=2
so variance would be
for p=m/N and
for very large value of N it would obviously
Zeta (Zipf) random variable
A discrete random variable is said to be Zeta if its probability mass function is given by
for the positive values of alpha.
In the similar way we can find the values of the expectation, variance and standard deviation.
In the similar way by using just the definition of the probability mass function and the mathematical expectation we can summarize the number of properties for the each of discrete random variable for example expected values of sums of random variables as
For random variables
In this article we mainly focused on some additional discrete random variable, its probability mass functions, distribution and the statistical parameters mean or expectation, standard deviation and variance, The brief introduction and simple example we discussed to give just the idea the detail study remains to discuss In the next articles we will move on continuous random variables and concepts related to continuous random variable ,if you want further reading then go through suggested link below. For more topics on mathematics, please this link.
Schaum’s Outlines of Probability and Statistics
- Conditional variance and predictions | Its Important Properties with 5+ Example
- Mathematical Expectation and random variable | Its 5 Important Properties
- Conditional Distribution | Its 5 Important Properties
- Jointly Distributed Random Variables | Its Important Properties & 5 Examples
- Gamma Distribution Exponential Family | Its 5 Important Properties