In this post, we explore the problem of finding the expected value of the minimum number of uniform random variables needed for their sum to exceed 1. We begin by presenting the problem statement, followed by two distinct solutions to the problem. The first solution calculates the expectation by definition, while the second solution approaches the problem by finding a differential equation for the expectation. Finally, we discuss an extension of the problem where we generalize the question to find the expected value of the minimum number of these variables needed for their sum to exceed an arbitrary value \(t\), where \(t \in [0, \infty]\).

Problem

Given \(X_1,...,X_n\) are iid random variables following a uniform distribution on the interval \([0,1]\), what is the expected value of the minimum number of these variables needed for their sum to exceed 1?

Solution 1: Using definition

Let \(N\) be the random variable representing the minimum number of uniform random variables needed for their sum to exceed 1, i.e.,

\[N = \inf_n \{ \sum_{i=1}^n X_i > 1\}.\]

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In this post, we discuss linear regression with grouped data and explore whether a certain reduction can be made when there are multiple responses given a covariate. The problem is formulated as follows:

Given data \(\left(y_{i j}, x_{i j}\right), 1 \leq j \leq n_i, 1 \leq i \leq m\), and we are asked to fit the model

\[y_{i j}=\alpha+\beta x_{i j}+\epsilon_{i j}\]

where \(\epsilon_{i j}\)’s are i.i.d. \(N\left(0, \sigma^2\right)\) and \(\sigma^2\) is known. The experiment is arranged so that

\[\qquad\qquad x_{i j}=x_i, \quad 1 \leq j \leq n_i, 1 \leq i \leq m\]

and the consequence of the model above is that

\[\text{E}[\bar{y}_i]=\beta_0+\beta_1 x_i, 1 \leq i \leq m,\quad \bar{y}_i=\sum_{j=1}^{n_i} y_{i j} / n_i .\]

1. Whether some of our time can be saved by using \(m\) pairs \(\left(\bar{y}_i, x_i\right)\) instead of the original data points.?

   Answer: If we know \(n_i\) of each group, we can use reduced pairs and get the same estimator as from the original data.

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