What is the relationship between two new random variables, Y1 and Y2, when defined in terms of i.i.d. Binomial random variables?
To understand the relationship between two new random variables Y1 and Y2, let's first discuss what it means for them to be defined in terms of i.i.d. (independent and identically distributed) Binomial random variables.
A binomial random variable represents the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success, denoted by p. It is characterized by two parameters: n, the number of trials, and p, the probability of success.
Now, if Y1 and Y2 are defined in terms of i.i.d. Binomial random variables, it means that they are derived from independent sets of Binomial random variables, and these random variables have the same parameters (n and p).
The relationship between Y1 and Y2 depends on how they are defined. Here are two possibilities:
1. Y1 and Y2 are sums of independent Binomial random variables: In this case, Y1 and Y2 represent the total number of successes in independent sets of Binomial trials. If Y1 is defined as the sum of n1 independent Binomial random variables, and Y2 is defined as the sum of n2 independent Binomial random variables, then the relationship between Y1 and Y2 can be analyzed using properties of sums of random variables, such as the laws of expectation and variance.
2. Y1 and Y2 are proportions of successes: In this case, Y1 and Y2 represent the proportion (or probability) of successes in independent sets of Binomial trials. If Y1 is defined as the number of successes divided by the total number of trials (n1), and Y2 is defined as the number of successes divided by the total number of trials (n2), then the relationship between Y1 and Y2 can be analyzed using properties of proportions, such as the sampling distribution of proportions and the Central Limit Theorem.
In summary, the relationship between two new random variables, Y1 and Y2, when defined in terms of i.i.d. Binomial random variables, depends on how they are defined (as sums or proportions) and can be further explored using relevant statistical properties and distributions.