Question

Let \(X\) be a random variable such that \(E\left(X^{2 m}\right)=(2 m) ! /\left(2^{m} m !\right), m=\) \(1,2,3, \ldots\) and \(E\left(X^{2 m-1}\right)=0, m=1,2,3, \ldots .\) Find the mgf and the pdf of \(X\).

Short answer

The moment generating function (MGF) of \(X\) is \(M_{X}(t) = \frac{1}{1 - t^2}\) for \(-1 < t < 1\) and the probability density function (PDF) is \(f_X(x) = \frac{1}{\pi}\frac{1}{1+x^2}\) for \(-\infty < x < \infty\)

Step-by-step solution

Derive the MGF

By definition, the Moment Generating Function (MGF) of a random variable \(X\) is \(M_X(t) = E[e^{tX}]\). By the power series definition of the exponential \(e^{r} = \sum_{m=0}^\infty \frac{r^m}{m!}\), it's seen that \(E\left[e^{tX}\right] = \sum_{m=0}^\infty \frac{t^m}{m!} E[X^m]\). Plug in the given moments to get the MGF. We have \(M_{X}(t) = E\left[e^{tX}\right] = \sum_{m=0}^\infty \frac{(t^2)^m}{m!} \frac{(2m)!}{2^m m!} = \sum_{m=0}^\infty \frac{(t^2)^m (2m)!}{2^m m! m!}\)

Simplify the MGF

By applying the definition of double factorial, we can simplify the MGF. We have \( M_{X}(t) = \sum_{m=0}^\infty \frac{(t^2)^m (2m)(2m-2)(2m-4) \dots 2}{2^m m! m!} = \sum_{m=0}^\infty (t^2)^m\), since all terms cancel out except \(t^2^m\)

Finalize the MGF

Finally, the summation leftover is a geometric series with ratio of \(t^2\). Hence, the Moment Generating Function becomes \(M_{X}(t) = \frac{1}{1 - t^2}\) for \(-1 < t < 1\)

Recognise the PDF form

The MGF corresponds to a Cauchy distribution, therefore the probability density function is \(f_X(x) = \frac{1}{\pi}\frac{1}{1+x^2}\) for \(-\infty < x < \infty\)

Key concepts

Cauchy Distribution

The Cauchy distribution is a continuous probability distribution that is known for its heavy tails and undefined mean and variance. It's a fascinating distribution because it doesn't follow the conventional rules that apply to other distributions like the normal distribution. The standard form of the Cauchy distribution is centered around its peak at zero, resembling a 'swell' defined by the probability density function (PDF) \( f(x) = \frac{1}{\pi(1 + x^2)} \).

• The central characteristic of the Cauchy distribution is that it does not have a defined mean or variance due to its heavy tails.
• This means its expected value, \(E(X)\), is undefined.
• This distribution is also known as the Lorentz distribution in physics.

Such properties make the Cauchy distribution an interesting subject of study where classical statistical theories don't always apply.

Probability Density Function (PDF)

The probability density function, commonly known as PDF, is a function that describes the likelihood of a random variable to take on a particular value. For continuous variables, the PDF allows us to calculate the probability that the variable falls within a particular range. The area under the curve of a PDF over a range gives us the probability of the variable falling within that range.
Key points about PDFs include:

• The PDF for a Cauchy distribution is given by \( f_X(x) = \frac{1}{\pi} \frac{1}{1+x^2} \).
• The total area under the PDF curve equals 1, fulfilling the condition of a probability function.
• Unlike a probability mass function (PMF) used for discrete random variables, a PDF is used for continuous random variables.

Understanding PDFs helps to interpret how probabilities are distributed across different potential values of a continuous variable.

Random Variable

A random variable is a fundamental concept in probability and statistics. It is essentially a variable whose values depend on outcomes of a random phenomenon. Random variables can be discrete, taking specific isolated values like counts, or continuous, covering a range like weights or heights.
Some important aspects of random variables include:

• They can be denoted with symbols such as \(X\) or \(Y\).
• For continuous random variables like in the Cauchy distribution, probabilities can only be calculated over intervals, not specific values.
• Functions like PDFs and cumulative distribution functions (CDFs) are used to describe their behavior.

Random variables serve as the building blocks for probability models used to infer and analyze real-world phenomena.

Expected Value

The expected value is a measure of the central tendency of a random variable. It can be thought of as the 'center of mass' of the probability distribution; it's where you would "expect" the value to be on average. It's determined through a weight sum of all possible values a random variable can take, with the weights being their probabilities.
Key points about expected values are:

• Expected value for a random variable \(X\) is generally noted as \(E(X)\).
• For a discrete random variable, \( E(X) = \sum x_i P(x_i) \).
• For continuous random variables, like in the Cauchy distribution, the expected value is an integral: \( E(X) = \int_{-\infty}^{\infty} x f(x) \, dx \).
• In the case of Cauchy distribution, the expected value, and indeed higher moments like variance, are undefined due to the way its tails behave.

The expected value provides a foundation for many statistical concepts, though it can be counterintuitive in certain distributions, such as the Cauchy distribution.