Geometric random variables
In this post, we will understand what are geometric random variables. To understand geometric random variables, it is required to understand what is a random variable and what is a binomial random variable.
Before going further let’s recall what are the conditions for Binomial Random Variable:
- Trials should be independent of each other.
- Each trial can be classified as either a success or a failure.
- Fixed number of trials.
- Probability of success on each trial should be constant.
Let’s try to understand geometric random variable with some examples. Consider two random variables X and Y defined as:
X = Number of sixes after 12 rolls of fair die
Y = Number of rolls until we get 6 on fair die
Can X be a binomial random variable?
Let’s list down all the properties of random variable X:
What is X - Number of sixes after 12 rolls of fair die
What is the experiment - You have to roll the die 12 times
What is the event/trial in experiment - Rolling of a die
Number of outcomes - 6
What are the outcomes - 1, 2, 3, 4, 5, 6
Is it a fair die - Yes
If it is not a fair die what is the probability of each outcome - NASo now we will ask whether each condition of binomial random variable is satisfied.
Are trials are independent?
Yes,Whenever we roll a die we know that outcome of that event is independent of any other trial. Can each trial be classified as either a success or a failure?
Yes,If the outcome is 6 then trial is success or else it is failure.Does this experiment has fixed number of trials?
Yes it has 12 trials.Whether the probability of success is constant during the experiment?
Yes the probability is constant.
Since all the conditions are satisfied, random variable X is a binomial random variable.
Can Y be a binomial random variable?
Let’s list down all the properties of random variable Y:
What is Y- Number of rolls until we get 6 on fair die.
What is the experiment - You have to roll the die until you get 6.
What is the event/trial in experiment - Rolling of a die
Number of outcomes - 6
What are the outcomes - 1, 2, 3, 4, 5, 6
Is it a fair die - Yes
If it is not a fair die what is the probability of each outcome - NASo now we will ask whether each condition of binomial random variable is satisfied.
Are trials are independent?
Yes,Whenever we roll a die we know that outcome of that trial is independent of any other trial.Can each trial be classified as either a success or a failure?
Yes,If the outcome is 6 then trial is success or else it is failure.Does this experiment has fixed number of trials?
No, We have to roll until we get 6Whether the probability of success is constant during the experiment?
Yes the probability is constant.
Random variable Y is a little bit different than X. From the above descriptions, we can see that random variable Y does not have a fixed number of trials. Apart from that, all other conditions are satisfied. This kind of variable is called geometric random variable.
From above example we can list down conditions for geometric random variable. Those are:
- Trials should be independent of each other.
- Each trial can be classified as either a success or a failure.
- how many trails until success
- Probability of success on each trial should be constant.
Conclusion:
In this post we understood what are geometric random variable and how these are different than binomial random variable.
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