Let f(x) and F(x) be the pdf and the cdf, respectively, of a...

Let $f(x)$ and $F(x)$ be the pdf and the cdf, respectively, of a distribution of the continuous type such that $f^{\prime}(x)$ exists for all $x$. Let the mean of the truncated distribution that has pdf $g(y)=f(y) / F(b),-\infty<y<b$, zero elsewhere, be equal to $-f(b) / F(b)$ for all real $b$. Prove that $f(x)$ is a pdf of a standard normal distribution.

Key Concepts

Stein’s Characterization of the Normal Distribution

Stein’s characterization provides a method to identify a normal distribution by leveraging relationships between differentiable functions of the random variable. Essentially, it involves showing that if a random variable satisfies a certain operational or differential identity, then it must be normally distributed. Such characterizations are used to demonstrate uniqueness and can simplify the analysis by reducing distributional properties to differential equations. In the problem, a condition involving the mean of a truncated distribution is used in a manner analogous to Stein’s method to conclude that the underlying distribution is the standard normal.

Differential Equations in Characterizing Distributions

Many probability distributions can be uniquely identified as solutions of specific differential equations that their cumulative or density functions satisfy. By establishing a differential equation based on given conditions—such as an expression for the truncated mean—one can often demonstrate that only a particular form of the pdf (for instance, the normal density) satisfies the equation under the appropriate boundary conditions. This approach is a powerful method in both theoretical and applied probability for characterizing and identifying distributions.

Truncated Distribution

A truncated distribution is a conditional probability distribution derived by restricting the support of an original distribution to a specified interval. When a distribution is truncated, its probability density function is adjusted (usually by a renormalization factor) so that the total probability over the new interval sums to one. This concept is vital when analyzing the behavior of a distribution within a limited range, especially when evaluating moments such as the mean under truncation.

Probability Density and Cumulative Distribution Functions

The probability density function (pdf) represents the likelihood of a random variable taking on a specific value, while the cumulative distribution function (cdf) gives the probability that the variable is less than or equal to a particular value. The cdf is obtained by integrating the pdf over the range up to that value. Understanding the relationship between these two functions is fundamental to studying continuous probability distributions and allows one to transition between local (pointwise) and cumulative characteristics of the distribution.

Differentiation under the Integral Sign (Leibniz’s Rule)

Differentiation under the integral sign, also known as Leibniz’s rule, is a technique that allows one to differentiate an integral with respect to a parameter that appears either in the limits of integration or the integrand. This method is essential when dealing with integrals that represent probabilities or expectations where the limits are variable. In the context of the problem, this tool is used to derive relationships involving derivatives of the cdf and pdf, which in turn lead to differential equations characterizing the distribution.

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