Let the density function of a random variable Y be given by...

Let the density function of a random variable $Y$ be given by
$$
f(y)=\left\{\begin{array}{ll}
\frac{2}{\pi\left(1+y^{2}\right)}, & -1 \leq y \leq 1 \\
0 & \text { elsewhere }
\end{array}\right.
$$
a. Find the distribution function.
b. Find $E(Y)$

Key Concepts

Symmetry

Symmetry in a probability distribution indicates that the pdf is mirrored around a central value, often making calculations like the expectation simpler. For symmetric distributions about zero, the contributions from positive and negative values cancel each other out, leading to an expected value of zero. Recognizing and utilizing symmetry is a key step in simplifying analyses and reducing computational effort when dealing with integration tasks in probability and statistics.

Expectation

The expectation, or mean, of a continuous random variable is given by the integral of the product of the variable and its pdf over the variable's support. It serves as a measure of the central tendency or the average outcome of the distribution. In many cases, properties such as symmetry in the pdf can simplify the calculation of the expectation, sometimes reducing it to zero if the distribution is symmetric about the origin.

Probability Density Function

A probability density function (pdf) is a function that describes the relative likelihood for a continuous random variable to take on a given value. In the context of the problem, the pdf is used to determine probabilities over intervals by integrating the function within those intervals. It is fundamental in defining the behavior of continuous random variables and ensuring that the total probability over the entire range is 1.

Cumulative Distribution Function

The cumulative distribution function (cdf) is derived from the pdf as the integral of the pdf from the lower bound of the variable's support up to a given value. It represents the probability that the random variable is less than or equal to a specific value. In problems like this, calculating the cdf involves integrating the pdf piecewise over its support, which aids in the assessment of probabilities and the formulation of other statistical measures.

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