Let X1, X2, ... be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to be the smallest k such that X1 + X2 + ... + Xx exceeds cn = 5 + 12/n, namely, Nn = min{k >1: X1 + X2 +...+Xk > cn} Does the limit lim P (NK>n) 100 exist? If yes, enter its numerical value. If not, enter -999.

Question

Let X1, X2, ... be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to be the smallest k such that X1 + X2 + ... + Xx exceeds cn = 5 + 12/n, namely, Nn = min{k >1: X1 + X2 +...+Xk > cn} Does the limit lim P (NK>n) 100 exist? If yes, enter its numerical value. If not, enter -999.

Answer 1

To determine whether the limit lim P(Nn > n) exists, we need to analyze the behavior of the sequence Nn.

First, let's break down the problem step by step:

1. Given a sequence of independent random variables Xi uniformly distributed on [0,1].
2. We define cn = 5 + 12/n.
3. Nn is defined as the smallest k such that X1 + X2 + ... + Xk exceeds cn.
Nn = min{k > 1: X1 + X2 + ... + Xk > cn}.

Now, we want to determine the limit lim P(Nn > n) as n approaches infinity.

To begin, let's calculate the expectation and variance of X1.

The uniform distribution on [0,1] has an expectation of μ = (b - a) / 2 = (1 - 0) / 2 = 1/2,
and a variance of σ^2 = (b - a)^2 / 12 = (1 - 0)^2 / 12 = 1/12.

For the sum of k independent random variables, the expectation is the sum of their individual expectations. Therefore, the sum of X1 + X2 + ... + Xk has an expectation of k/2.

Now, we aim to use the Central Limit Theorem (CLT) to understand the behavior of the sum X1 + X2 + ... + Xk. The CLT states that the sum of a large number of independent and identically distributed random variables, each with finite mean and variance, tends towards a normal distribution as the sample size increases.

Since each Xi has an expectation of 1/2 and a variance of 1/12, the sum X1 + X2 + ... + Xk has an expectation of (k/2) and a variance of (k/12).

Now, let's consider the behavior of Nn. For a given n, we are interested in finding the smallest k such that X1 + X2 + ... + Xk > cn.

Using the Central Limit Theorem, we can approximate X1 + X2 + ... + Xk as a normally distributed random variable with mean (k/2) and variance (k/12) as k increases.

Since cn = 5 + 12/n, we can rewrite the condition X1 + X2 + ... + Xk > cn as (X1 + X2 + ... + Xk - (k/2)) / sqrt(k/12) > (cn - (k/2)) / sqrt(k/12).

However, as n approaches infinity, cn approaches 5, and (cn - (k/2)) / sqrt(k/12) approaches negative infinity. Hence, P(Nn > n) becomes P(X > -∞), which is equal to 1.

Therefore, the limit lim P(Nn > n) as n approaches infinity is equal to 1.