# Questions tagged [ratio]

the quantitative relation between two amounts showing the number of times one value contains or is contained within the other. Mathematically the quotient of one amount by another amount.

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48k views

### Expected number of ratio of girls vs boys birth

I have came across a question in job interview aptitude test for critical thinking. It is goes something like this: The Zorganian Republic has some very strange customs. Couples only wish to have ...
2k views

### I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
26k views

### How to compute the confidence interval of the ratio of two normal means

I want to derive the limits for the $100(1-\alpha)\%$ confidence interval for the ratio of two means. Suppose, $X_1 \sim N(\theta_1, \sigma^2)$ and $X_2 \sim N(\theta_2, \sigma^2)$ being independent, ...
2k views

### Ratios in Regression, aka Questions on Kronmal

Recently, randomly browsing questions triggered a memory of on off-hand comment from one of my professors a few years back warning about the usage of ratios in regression models. So I started reading ...
571 views

### An unbiased estimator of the ratio of two regression coefficients?

Suppose you fit a linear/logistic regression $g(y) = a_0 + a_1\cdot x_1 + a_2\cdot x_2$, with the aim of an unbiased estimate of $\frac{a_1}{a_2}$. You are very confident that both $a_1$ and $a_2$ ...
16k views

### What are the issues with using percentage outcome in linear regression?

I have a study where many outcomes are represented like percentages and I'm using multiple linear regressions to asses the effect of some categorical variables on these outcomes. I was wondering, ...
3k views

### Should types of data (nominal/ordinal/interval/ratio) really be considered types of variables?

So for instance here are the definitions that I get from standard text books Variable - characteristic of population or sample. ex. Price of a stock or grade on a test Data - actual ...
2k views

### What is the ratio of uniform and normal distribution?

Let $X$ follow a uniform distribution and $Y$ follow a normal distribution. What can be said about $\frac X Y$? Is there a distribution for it? I found the ratio of two normals with mean zero is ...
4k views

### Distribution of the ratio of dependent chi-square random variables

Assume that $X = X_1 + X_2+\cdots+ X_n$ where $X_i \sim N(0,\sigma^2)$ are independent. My question is, what distribution does $$Z = \frac{X^2}{X_1^2 + X_2^2 + \cdots + X_n^2}$$ follow? I know ...
1k views

### How to find the sample points that have statistically meaningful large outlier ratios between two values of the point?

As an example application, consider following two properties of Stack Overflow users: reputation and profile view counts. It is expected that for most users those two values will be proportional: high ...
350 views

### Expected value of maximum ratio of n iid normal variables

Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$. How would one be able to upper bound the expected maximum of the ratio ...
6k views

### Testing Sharpe Ratio significance

What is the proper way to test the significance of Sharpe Ratios or Information Ratios? The Sharpe Ratios will be based on various equity indices and may have variable look-back periods. One solution ...
1k views

### If X/Y has the same distribution as Z, is it true that X has the same distribution as YZ?

Let X, Y and Z be three independent random variables. If X/Y has the same distribution as Z, is it true that X has the same distribution as YZ?
1k views

### Expected value of ratio of correlated random variables?

For independent random variables $\alpha$ and $\beta$, is there a closed form expression for $\mathbb E \left[ \frac{\alpha}{\sqrt{\alpha^2 + \beta^2}} \right]$ in terms of the expected values and ...
838 views

### How to correctly analyze fatality rate and daily deaths of Chinese and Italian COVID-19 outbreak?

This is a strange case of difference in fatality rate between Chinese and Italian covid-19 outbreak. In my knowledge, fatality rate is a ratio between deaths from a certain disease compared to the ...
206 views

### Do random variables follow the same algebraic rules as ordinary numbers?

In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim ...
14k views

### Is the ratio distribution of two normally distributed variables ever normal?

Let $Z = X / Y$ where $X$ and $Y$ are normally distributed random variables. Is $Z$ normally distributed for any $X$ and $Y$?
961 views

### If my goal is to test the absolute change of the ratios, can I compare the ratios directly without log transformation?

Ratios (e.g. $Z$=$Y$/$X$) are frequently used (e.g. fold-changes in mRNA or protein expression, body mass index [BMI], etc.). Many people advise that variables coded as ratios (e.g. fold-change) ...
3k views

### Using Fieller's theorem to calculate the confidence interval of a ratio (paired measurements)

If you have two means (with their own confidence intervals) and want to represent them as a ratio, how do calculate the confidence interval for the ratio? An answer that was given to me, mentions ...
516 views

### Ratio of correlated sample variances (gamma distributed)

for $N$ samples of two correlated random variables $X \sim N\left(0,\sigma_X^2\right)$ and $Y \sim N\left(0, \sigma_Y^2\right)$ with correlation $\rho$, I am analyzing the ratio of the sample ...
703 views

### How should I evaluate the expectation of the ratio of two random variables?

Let $A$ and $B$ be random variables and $f(A,B)=\frac{A}{B}$. How should I approximate $E(f(A,B))$? I think a Taylor expansion may be in order, but I am not sure how to fire it off in this function. ...
1k views

### Analysing ratios of variables

Ratios (X/Y, e.g. body mass index) are variables with odd distributions. They have no means or moments (e.g. variances, skewness or kurtosis). Thus, my questions are: How to compare between (or among)...
331 views

### Direct way of calculating $\mathbb{E} \left[ \frac{\textbf{h}^{H} \textbf{y}\textbf{y}^{H} \textbf{h}}{ \| \textbf{y} \|^{4} } \right]$

Considering the following random vectors: \begin{align} \textbf{h} &= [h_{1}, h_{2}, \ldots, h_{M}]^{T} \sim \mathcal{CN}\left(\textbf{0}_{M},d\textbf{I}_{M \times M}\right), \\[8pt] \textbf{w} &...
190 views

### Compare two set of ratios [closed]

I'm looking for a statistical test to compare two set of ratios. For example A = [(2,13), (10,50), (23, 86), (33, 133)] B = [(6,10), (21,33), (30,62), (44,80)] ...
917 views

### Why would I use ratio estimation instead of regression estimation to estimate means?

I am taking a graduate course on survey data analysis. I was recently introduced to ratio estimation and regression estimation. I understand that using ratio estimator may be easier if we are ...
906 views

### What is the distribution of the ratio between independent Beta and Gamma random variables?

What would be the distribution of the following equation: $$y = \frac{a}{(a+d)^2}$$ where $a, d$ $\sim$ $\Gamma(M,c)$. Additionally, $a$ and $d$ are independent random variables.
773 views

### median(a)/median(b) not equal median(a/b)

I'm sure this is a very straightforward question but it came up in my work today and I could not think of the reasoning behind it. I had two sets of numeric values (A & B) and was looking at the ...
161 views

### Distribution of the ratio of a Normal distribution divided by Lognormal distribution

I want to know the distribution (and the moments) of a variable, $Z = X/Y$, where $X\sim \mathcal{N}(\mu_{x}, \sigma^{2}_{x})$, and , $Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$? Hence, what I want ...
109 views

### How to find the expectation $\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right]$?

Given the independent and complex Gaussian random variables $h$ and $w$, how does one can find the following expectation? \mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right] = \int_{\mathbb{C}}\...
161 views

### Can a ratio of random variables be normal? [duplicate]

For a pair of random variables $Y$ and $Z$, is it possible that their ratio $X:=\frac{Y}{Z}$ is (exactly, not asymptotically) normally distributed? If so, could you offer an example of the ...
2k views

### Beta regression and regression diagnostics. Do we need to check for normality and other diagnostics?

I have a dependent variable which is a ratio and 0 < y < 1 condition holds. I will apply betareg in Stata but I am not sure what are the diagnostics that are ...
342 views

### Disentangling the effect of a ratio from the effects of the numerator & denominator in linear models

My question: What is the most appropriate way to analyse cases in which two variables and their ratio are all believed to influence some outcome? I would ideally like guidance in a regression ...
227 views

### Generating identically-distributed random variables with a constraint

Is there a way to generate identically-distributed random variables (eg $x_1,x_2,x_3,x_4$) with the following constraint: $\frac{x_1*x_2}{x_3*x_4} ≡ 1$ $x \in (0,1)$ Please note that simply ...
940 views

### Uncertainty in a fractional count

What is the uncertainty (68% confidence level) of $N/M$, where $N$ is the number of entries that pass a cut and $M$ is the total number of entries? ($N$ and $M$ are both integers, and I'm interested ...
183 views

1k views

### Bias of estimating a Ratio?

I would like to know why 1/N * Summation of(y/x) is a worse estimator than average of y divided by average of x. This is in the context of constructing an estimator of a ratio where the ratio would ...
191 views

### What is the distribution of $x/(x + y)$ for $x$, $y$ independent and normally distributed with mean $0$

What is the distribution of $x/(x + y)$ for independent and normally distributed $x$, $y$ with mean $0$? I'm aware $x/y$ has a Cauchy distribution but I don't know if there is a way to make use of it. ...
334 views

### Ratio that accounts for different sample sizes

I'm trying to get a measure for a set of data that indicates someone's "spam score". Essentially, the higher the spam score, the more likely they are to be spammers. Right now, I'm measuring a person'...
140 views

### Distribution of *conditional* frequencies when frequencies follow a Dirichlet distribution

Context: we have a large number of individuals characterized by two binary traits; call these $T$ with values $\{0,1\}$, and $T'$ with values $\{0',1'\}$. So there are four types of individuals: $00'$,...
198 views

### One of the independent variables in a logistic regression is the ratio of other two inputs. Is this okay?

Suppose that I have a logistic regression with continuous independent variables $a$,$b$, and $c$. In my logistic regression, $c = \frac{a}{b}$. Is it all right to include variable $c$ in the ...
246 views

### Judging the quality of a statistical model for a percentage

I have a data set with multiple predictors, and a single response variable which is a percentage, thus is bounded between 0 and 100. I cannot share the dataset unfortunately. I would like to build a ...
4k views

### How to (properly) analyze the sex ratio

Few weeks ago, I was listening to a "HowToDoScience" lecture. In the section which focused on appropriate statistics in articles the lecturer said: Person who will analyse the sex ratio by the chi-...
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### T-test or Z-test of two proportions?

We did an intervention to reduce the frequency of use of a certain type of treatment and now I want to compare the data pre vs post intervention. For each month I have the number of treatments per ...