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Chapter 3: Problem 12

Show that the \(t\) -distribution with \(r=1\) degree of freedom and the Cauchy distribution are the same.

Short Answer

Expert verified
The t-distribution with 1 degree of freedom and the Cauchy distribution are the same.

Step by step solution

01

Write down the formula for t-distribution

The probability density function of t-distribution is given by\n\[ p(x) = \frac{(1+x^{2}/r)^{- (r+1)/2}}{B(r/2, 1/2)}, \]\nwhere B is the beta function, x is the random variable, and r is the degree of freedom.
02

Write down the formula for Cauchy distribution

The probability density function of Cauchy distribution is given by\n\[ p(x) = \frac{1}{蟺(1+x^{2})},\] \nwhere x is the random variable.
03

Plug r = 1 into the t-distribution formula

In this case, t-distribution transforms into\n\[ p(x) = \frac{1}{B(1/2, 1/2)(1+x^{2})}, \] \nNote B(1/2, 1/2) = 蟺.
04

Compare the two distributions

From step 3, as \(B(1/2, 1/2) = 蟺\), we have\n\[ p(x) = \frac{1}{蟺(1+x^{2})}. \] \nWe can observe that, the two distributions ie., t-distribution (with r=1) and Cauchy distribution are identical.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution. It is used primarily in hypothesis testing and confidence interval estimation for small sample sizes or when the population standard deviation is unknown. A key feature of the t-distribution is that it has heavier tails compared to the normal distribution, which means it can better capture the variability expected in smaller samples. The shape of the t-distribution depends on the degrees of freedom (r), which is derived from sample size.
  • As the number of degrees of freedom increases, the t-distribution approaches the normal distribution.
  • A t-distribution with lower degrees of freedom will have thicker tails, implying more data variability.
  • At one degree of freedom (r=1), the t-distribution is coincidentally the same as the Cauchy distribution.
Understanding the t-distribution is crucial for interpreting results in statistical analysis, particularly when dealing with small sample sizes.
Cauchy distribution
The Cauchy distribution is another type of probability distribution known for its peculiar properties. Sometimes called a Lorentz distribution, it is unique because it does not have a defined mean or variance. This is because the distribution has very heavy tails, which results in undefined moments.
  • The Cauchy distribution is symmetric around its peak, similar to both the normal and t-distributions.
  • It arises naturally when considering ratios of standard normal variables.
  • This distribution is used in applications like physics, specifically in resonance behavior and other phenomena.
Despite sharing a similar appearance to the normal distribution, the Cauchy distribution does not follow the same statistical properties, making it unsuitable for typical parameter estimation techniques like mean and variance.
degree of freedom
Degrees of freedom (commonly abbreviated as df) is a statistical term that represents the number of independent values or quantities which can be assigned to a statistical distribution. When performing statistical analyses involving distributions like the t-distribution, degrees of freedom help determine the shape of the distribution.
  • The degrees of freedom are often related to the sample size or the number of independent variables in the dataset.
  • In the context of the t-distribution, the degrees of freedom are calculated as the sample size minus one (n-1), reflecting the variability in a sample.
  • The greater the degrees of freedom, the closer the t-distribution resembles a normal distribution.
In essence, degrees of freedom are crucial in understanding the variance and shape of distributions in statistical analysis, helping guide appropriate usage in hypothesis testing.

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Most popular questions from this chapter

Let the pmf \(p(x)\) be positive on and only on the nonnegative integers. Given that \(p(x)=(4 / x) p(x-1), x=1,2,3, \ldots\), find the formula for \(p(x)\). Hint: Note that \(p(1)=4 p(0), p(2)=\left(4^{2} / 2 !\right) p(0)\), and so on. That is, find each \(p(x)\) in terms of \(p(0)\) and then determine \(p(0)\) from $$ 1=p(0)+p(1)+p(2)+\cdots $$

Let \(X\) have a geometric distribution. Show that $$ P(X \geq k+j \mid X \geq k)=P(X \geq j) $$ where \(k\) and \(j\) are nonnegative integers. Note that we sometimes say in this situation that \(X\) is memoryless.

The approximation discussed in Exercise \(3.2 .8\) can be made precise in the following way. Suppose \(X_{n}\) is binomial with the parameters \(n\) and \(p=\lambda / n\), for a given \(\lambda>0 .\) Let \(Y\) be Poisson with mean \(\lambda\). Show that \(P\left(X_{n}=k\right) \rightarrow P(Y=k)\), as \(n \rightarrow \infty\), for an arbitrary but fixed value of \(k\). Hint: First show that: $$ P\left(X_{n}=k\right)=\frac{\lambda^{k}}{k !}\left[\frac{n(n-1) \cdots(n-k+1)}{n^{k}}\left(1-\frac{\lambda}{n}\right)^{-k}\right]\left(1-\frac{\lambda}{n}\right)^{n} $$

If the random variable \(X\) has a Poisson distribution such that \(P(X=1)=\) \(P(X=2)\), find \(P(X=4)\).

Let \(Y\) be the number of successes throughout \(n\) independent repetitions of a random experiment with probability of success \(p=\frac{1}{4}\). Determine the smallest value of \(n\) so that \(P(1 \leq Y) \geq 0.70\)
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