For each positive integer n, define fn to be the normal...

For each positive integer $n$, define $f_n$ to be the normal density with mean zero and variance $n$, i.e., $$ f_n(x)=\frac{1}{\sqrt{2 n \pi}} e^{-\frac{x^2}{2 n}} . $$ (i) What is the function $f(x)=\lim _{n \rightarrow \infty} f_n(x)$ ? (ii) What is $\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x$ ? (iii) Note that $$ \lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x \neq \int_{-\infty}^{\infty} f(x) d x . $$ Explain why this does not violate the Monotone Convergence Theorem, Theorem 1.4.5.

For each positive integer $n$, define $f_n$ to be the normal density with mean zero and variance $n$, i.e.,
$$
f_n(x)=\frac{1}{\sqrt{2 n \pi}} e^{-\frac{x^2}{2 n}} .
$$
(i) What is the function $f(x)=\lim _{n \rightarrow \infty} f_n(x)$ ?
(ii) What is $\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x$ ?
(iii) Note that
$$
\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x \neq \int_{-\infty}^{\infty} f(x) d x .
$$

Explain why this does not violate the Monotone Convergence Theorem, Theorem 1.4.5.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric bell-curve shape and defined by its mean and variance. In this problem, each function f? is a normal density with mean zero and variance n, meaning that as n increases, the distribution becomes more spread out and its peak height decreases, although the total area under the curve remains 1.

Pointwise Convergence

Pointwise convergence concerns the behavior of a sequence of functions at each individual point in the domain. Here, for any fixed x, the value f?(x) tends to 0 as n tends to infinity, because the density’s spread causes the function values at any specific point to diminish, even though the overall mass (area under the curve) remains constant.

Integration of Probability Densities

The integral of a probability density function over the entire real line represents the total probability, which is always 1 for valid probability distributions. Although the pointwise limit of each f?(x) is 0, leading to the misconception that the integral should vanish, the property of probability densities is ensured by the way the 'mass' is distributed as the variance grows, keeping the total integral equal to 1.

Monotone Convergence Theorem

The Monotone Convergence Theorem (MCT) applies to sequences of nonnegative measurable functions that increase monotonically (i.e., f?(x) ? f???(x) for all n and almost every x). In this example, the sequence {f?} does not satisfy the monotonicity condition because the functions are not ordered in an increasing or decreasing fashion with respect to n. Therefore, the apparent discrepancy between the limit of the integrals and the integral of the limit does not contradict the MCT, as the theorem’s hypotheses are not applicable here.

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