Limit Theorems in Probability Theory

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What do limit theorems in probability theory typically investigate?

Properties of a function of a sequence of random variables as the sequence length approaches infinity (correct)

Properties of a sequence of random variables as the sequence length remains constant

Properties of a single random variable as its value approaches infinity

Properties of a function of a sequence of non-random variables as the sequence length approaches infinity

What practical benefit do limit theorems provide?

They allow us to compute exact quantities without approximation

They provide a way to compute the exact mean and variance of any random variable

They allow us to use limits as approximations for quantities that are difficult to compute exactly (correct)

They provide a way to compute the exact distribution of any random variable

In the example given, what does the sample mean $\overline{X}$ represent?

The average of the random variables $X_1, X_2, \ldots, X_n$ (correct)

The variance of the random variables $X_1, X_2, \ldots, X_n$

The limit of the random variables $X_1, X_2, \ldots, X_n$ as $n \rightarrow \infty$

The expected value of the random variables $X_1, X_2, \ldots, X_n$

If the random variables $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) with mean $\mu$ and variance $\sigma^2$, what is the expected value of the sample mean $\overline{X}$?

$\mu$ Signup and view all the answers

If the random variables $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) with mean $\mu$ and variance $\sigma^2$, what is the variance of the sample mean $\overline{X}$?

$\frac{\sigma^2}{n}$ Signup and view all the answers

What does the fact that the variance of the sample mean $\overline{X}$ decreases as $n$ increases indicate?

$\overline{X}$ is likely to be close to $\mu$ for large values of $n$ Signup and view all the answers

What does it mean for a sequence of real numbers $a_n$ to converge to a real number $a?

For any $\varepsilon > 0$, we can always make $|a_n - a| < \varepsilon$ if $n$ is large enough. Signup and view all the answers

Why can't we say that the sample mean $\bar{X}$ converges to the population mean $\mu$ in the same way that a sequence of real numbers converges to a real number?

Because we can never guarantee that $|\bar{X} - \mu| < \varepsilon$ for some $\varepsilon > 0$ and large enough $n$. Signup and view all the answers

What is the main idea behind the law of large numbers?

The probability that $|\bar{X} - \mu| > \varepsilon$ goes to 0 as $n$ increases, for any $\varepsilon > 0$. Signup and view all the answers

What is the key assumption needed for the proof of the law of large numbers presented in the text?

The random variables $X_i$ must have finite variance $\sigma^2. Signup and view all the answers

What is the meaning of the notation $\bar{X} \xrightarrow{P} $ as $n \to \infty?

The probability that $|\bar{X} - \mu| > \varepsilon$ goes to 0 as $n$ increases, for any $\varepsilon > 0. Signup and view all the answers

Limit Theorems in Probability Theory

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Questions and Answers

What do limit theorems in probability theory typically investigate?

What practical benefit do limit theorems provide?

In the example given, what does the sample mean $\overline{X}$ represent?

If the random variables $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) with mean $\mu$ and variance $\sigma^2$, what is the expected value of the sample mean $\overline{X}$?

If the random variables $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) with mean $\mu$ and variance $\sigma^2$, what is the variance of the sample mean $\overline{X}$?

What does the fact that the variance of the sample mean $\overline{X}$ decreases as $n$ increases indicate?

What does it mean for a sequence of real numbers $a_n$ to converge to a real number $a?

Why can't we say that the sample mean $\bar{X}$ converges to the population mean $\mu$ in the same way that a sequence of real numbers converges to a real number?

What is the main idea behind the law of large numbers?

What is the key assumption needed for the proof of the law of large numbers presented in the text?

What is the meaning of the notation $\bar{X} \xrightarrow{P} $ as $n \to \infty?

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