What is the distribution of the sum of n iid Bernoulli random variables?

So ##\sum_{i=1}^n Y_i## is a sum of Bernoulli random variables, which is a binomial distribution with parameters ##n## and ##p##.In summary, the distribution of ∑ni=1Yi is a binomial distribution with parameters n and p, where p is the probability that Xi ≤ μ. This can be calculated using the fact that each Yi is a Bernoulli random variable with probability p.
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Qyzren
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Homework Statement


Let X1, X2, ..., Xn be iid random variables with continuous CDF FX and suppose the common mean is E(Xi) = μ. Define random variables Y1, Y2, ..., Yn by
Yi = 1 if Xi > μ; 0 if Xi ≤ μ. Find the distribution of ∑ni=1Yi.

I'm having a hard time figuring out how to begin to find the distribution.

Homework Equations


Possibly Yn = (∑Xi - nμ ) /√n σ ?

The Attempt at a Solution


I'm having a hard time knowing where to begin...
If the question was p instead of μ, then Xi ~ Bernoulli(p) and the sum of n iid Bernoulli(p) is Binomial(n, p).

But since we have μ and not p, I'm also thinking that the central limit theorem tells us the sum of any random variable is always yields a normal distribution, with mean nμ? and variance ...?

Any help to get me in the right direction would be greatly appreciated!
 
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  • #2
Define ##p_i=Prob(X_i\le \mu)##. Since the ##X_i## are iid, ##p_i## is the same for all ##i## so we can just write it as ##p##. Then each ##Y_i## is a Bernoulli.
 
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Related to What is the distribution of the sum of n iid Bernoulli random variables?

1. What is a distribution?

A distribution is a way of organizing and representing a set of data. It shows how frequently each value or range of values occurs in a given dataset, and can help us understand the overall pattern and characteristics of the data.

2. Why is it important to find the distribution of data?

Finding the distribution can help us gain insights and make informed decisions about the data. It can help us identify trends, outliers, and relationships between variables, and can also serve as a basis for statistical analysis and modeling.

3. How do you determine the distribution of data?

There are several methods for determining the distribution of data, such as visual methods (e.g. histograms, box plots) and statistical tests (e.g. normality tests, chi-square tests). The choice of method depends on the type of data and the research question being addressed.

4. What are some common types of distributions?

Some common types of distributions include normal (bell-shaped), uniform, bimodal, and skewed (positively or negatively). The type of distribution can provide information about the underlying data and may influence the choice of statistical analyses.

5. How does the shape of a distribution affect the interpretation of the data?

The shape of a distribution can provide information about the central tendency and variability of the data. For example, a normal distribution with a symmetric shape indicates that the majority of the data is clustered around the mean, while a skewed distribution may suggest the presence of outliers or a non-linear relationship between variables.

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