Linked Questions

3
votes
0answers
3k views

why is the variance of t-distribution with 1 and 2 degrees of freedom undefined while these distributions can be drawn? [duplicate]

The variance of a t-distribution is given by df/(df-2), hence the t-distribution with 1 and 2 degrees of freedom have no defined variance. Yet these distributions do exist and can be drawn, so one ...
0
votes
0answers
53 views

Expected value $=\infty$? [duplicate]

If we let $U_1, U_2, U_3,..., U_n$ be uniform (0,1), find $$\mathbb E[\sum_{i=0}^n iU_i^{i-1}]$$which, using the linearity of expectation, gives $$\sum_{i=0}^n \mathbb E[i U_i^{i-1}]$$ Doing this ...
0
votes
0answers
49 views

Does expectation of a r.v. always converge? [duplicate]

Let X be a random variable. Will its expectation E[X] expressed as a (possibly infinite) sum converge for arbitrary X? My intuition would be yes. Since E[X] is a function of a r.v., E[X] is a r.v. ...
41
votes
4answers
32k views

What is the difference between finite and infinite variance

What is the difference between finite and infinite variance ? My stats knowledge is rather basic; Wikipedia / Google wasn't much help here.
23
votes
6answers
7k views

What makes the mean of some distributions undefined?

Many PDFs range from minus to positive infinity, yet some means are defined and some are not. What common trait makes some computable?
19
votes
2answers
3k views

Why is the Cauchy Distribution so useful?

Could anyone give me some practical examples of the Cauchy Distribution? What makes it so popular?
17
votes
3answers
8k views

Does non-zero correlation imply dependence?

We know of the fact that zero correlation does not imply independence. I am interested in whether a non-zero correlation implies dependence - i.e. if $\text{Corr}(X,Y)\ne0$ for some random variables $...
9
votes
4answers
14k views

How does the expected value relate to mean, median, etc. in a non-normal distribution?

How does the expected value of a continuous random variable relate to its arithmetic mean, median, etc. in a non-normal distribution (eg. skew-normal)? I'm interested in any common/interesting ...
7
votes
2answers
2k views

WLLN: can expectation exist but be infinite?

Weak law of large numbers: Let $\{h_i, i = 1, \dots n\}$ be an $m \times q$ sequence of iid random variables with mean $\mu = E[h_i]$ that exists and is finite. Then $1/n \sum_{i = 1}^n h_i \...
3
votes
1answer
24k views

What is the Mean and Standard Deviation of the division of two random variables? [duplicate]

I have two normally-distributed independent random variables X and Y and I need to calculate its division Z. As far as I understand the mean of Z is $\mu_Z = \frac{\mu_X}{\mu_Y}$, but I don't know ...
9
votes
2answers
807 views

Bayesian lighthouse location estimation

I am trying to learn Stan in R and as a fun challenge I am trying to estimate the location of a lighthouse based on the observed flashes. But the models I tried do not converge (Rhat != 1) or have ...
9
votes
1answer
3k views

Central Moments of Symmetric Distributions

I am trying to show that the central moment of a symmetric distribution: $${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf E[(...
5
votes
1answer
7k views

When does a distribution not have a mean or a variance?

I believe I read today a phrase which went something like this: If a distribution has a mean and a variance ... So I guess that means some distributions do not have means or variances? I fiend ...
7
votes
1answer
4k views

What are the mean and variance of the ratio of two normal variables, with non-zero means?

If X,Y are normal independent N(a,s), N(b,s') what are means and variances of the ratio X/Y ?
3
votes
3answers
397 views

Where does the expected value definition come from? [duplicate]

The definition of the expected value on the domain $[a,b]$ is given by $$E[X] := \int_a^b x f(x) \, \mathrm dx $$ I understand what the mean is, but I don't fully understand how this specific equation ...
5
votes
2answers
334 views

What is the expected value of $\frac{X}{X+Y}$?

I am trying to find the expected value of $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg]$. I started with writing $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg] = E\Bigg[X\cdot\frac{1}{X+Y}\...
5
votes
2answers
2k views

Strictly Stationary Time Series with Infinite Moments

Can someone give me an example of a strictly stationary time series with infinite moments? I am reading a book on Time Series by Wayne A. Fuller where it is said that a strictly stationary time series ...
3
votes
2answers
338 views

Find a general normal stationary process

I am wondering how to find the general normal stationary process satisfying $X_{n+2} + X_{n} = 0$. Any help would be much appreciated, although I am relatively new to this space so some details how to ...
4
votes
2answers
1k views

Expectation of quotient of linear combinations of independent standard normal random variables

Let $a, b, c, d, e, f$ be complex numbers with nonnegative real parts and nonnegative imaginary parts, and let $X_{1}, X_{2}, X_{3}, X_{4}$ be independent standard normal random variables. How can I ...
4
votes
1answer
731 views

When can't a confidence interval be constructed?

In one of my econometrics assignments, we were asked to consider the effect of measurement error in the dependent variable of a simple linear regression. And I was just wondering, under what ...
2
votes
1answer
840 views

Does the k-th moment exists when $E[X^k]$ is infinite in ether (one) positive or negative direction? [duplicate]

I'm reading Probability and Statistics by DeGroot and Schervish, and in it it said that the expectation of a random variable exists if at least one of the integrals over all negative/positive values ...
4
votes
1answer
103 views

Fisher's example of a non-converging mean for N towards infinity

In one of Fisher's classical paper [1] I stumbled over the following: If the frequency with which the variate $x$ falls into the range $dx$, be given by $$df = \frac{1}{\pi}\frac{dx}{1+(x-m)^2}$$ ...
1
vote
1answer
117 views

Why do I get worse regression metrics when I add more instances to the problem?

I find this counter-intuitive. First I chose randomly 7000 instances and my model explains 55% of the variance. Then I train with the whole dataset (43000) and I get negative $R^2$. How is this even ...
1
vote
1answer
114 views

Population vs sample

When we think of linear regression, the implicit assumption is that we only observe a small fraction of a possibly infinite large population. Thinking of simple averages, imagine a fair die. The ...
2
votes
0answers
58 views

A question regarding symmetry properties of a uniform distribution [duplicate]

Was anyone able to explain why $$E(U_2) = 0$$ I don't quite understand what the relevance of the underlined statement - "by the symmetry of $U_1$" in determining $E(U_2)$ is edit: I get it now, ...
0
votes
0answers
49 views

A practical example where maximum likelihood correctly estimates an underlying parameter, but where least squares would fail?

Forgive my very limited understanding. I am trying to learn about maximum likelihood estimation, and how it differs from least-squares estimation. From reading a little, I understand that the two are ...
0
votes
0answers
34 views

Proof that ratio of normal distributions is Cauchy [duplicate]

I recall that the ratio of 2 independent standard normal random variables follows a Cauchy distribution, but I don't recall if there is a proof for this.
0
votes
0answers
22 views

Why Cauchy distribution has no Moment-generating Function? [duplicate]

I'm reading Casella and Berger's book. In chapter 5, they show that we can recover the distribution of the sample mean by using the following relationship between the MGF of the underlying random ...