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How to calculate the variance of the ratio of random variables?
The ratio of two random variables does not in general have a well-defined variance, even when the numerator and denominator do. A simple example is the Cauchy distribution which is the ratio of two independent normal random variables. As Sivaram has pointed out in the comments, the formula you have given…
How is the sample ratio of a sample estimated?
The sample ratio ( r) is estimated from the sample That the ratio is biased can be shown with Jensen’s inequality as follows (assuming independence between x and y): Under simple random sampling the bias is of the order O ( n−1 ).
How is the variance of the f ratio calculated?
To calculate the F ratio, two estimates of the variance are made. Variance between samples: An estimate of σ2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.). If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes.
How are ratio estimators biased to the expected value?
The ratio estimators are biased. The bias occurs in ratio estimation because E(y=x) =E(y)=E(x) (i.e., the expected value of the ratio6= the ratio of the expected values.6
How to calculate the ratio of two normal variables?
For real α and β > 0, suppose Cauchy(α, β) denotes the density f(x) = β π ( ( x − α)2 + β2), x ∈ R. It can be shown using a change of variables or otherwise that if (X, Y) has a standard bivariate normal distribution with zero means, unit variances and correlation ρ, then X Y has a Cauchy(ρ, √1 − ρ2) distribution.
Which is the ratio of two Gaussian variables?
The random variable associated with this distribution comes about as the ratio of two Gaussian (normal) distributed variables with zero mean. Thus the Cauchy distribution is also called the normal ratio distribution.
Is the mean equal to the product of independent variables?
In the Product distribution section, and derived from Mellin transform theory (see section above), it is found that the mean of a product of independent variables is equal to the product of their means. In the case of ratios, we have It has been suggested that this section be split out into another article titled Normal ratio distributions.