# Distribution of ratio std Normal and sqrt chi squared

• I
• WWGD
In summary: So the square root of a squared normal is not a normal, but it is half of one.In summary, the ratio of a standard normal by the square root of a chi-squared distribution is a t-distribution. The square root of a chi-squared distribution with k degrees of freedom is the chi distribution with k degrees of freedom. However, the square root of a squared normal is not a normal, but it is half of one. This is because the square root is not an injection or bijection, but rather a composition with absolute value.

#### WWGD

Gold Member
Hi all,
I am trying to understand two things from a paper
The ratio of a standard normal by the square root of a a chi squared divided by its df ( degrees of freedom) is a t distribution. So
1) What is the dist of square root of Chi squared? I know a normal squared is a chi squared, but a chi squared may not necessarily come about ad the square of a normal
2) Why does the ratio of a standard normal by the square root of a chi squared a t distribution? What result is this?
Only somewhat related result can think of is that ratio of independent standard normals ( of course, nonzero denominator) is a Cauchy. Edit: I wanted to double check the claim that the square root of a chi squared is a chi squared because this does not seem true about the square root of a square normal, which seems should be normal.
Thanks.

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The distribution of a random variable that is the square root of a chi-squared distribution with k degrees of freedom is the chi distribution with k degrees of freedom. Details are here.
WWGD said:
: I wanted to double check the claim that the square root of a chi squared is a chi squared
the claim is incorrect. It's not another chi-squared, it's a chi.
WWGD said:
2) Why does the ratio of a standard normal by the square root of a chi squared a t distribution?
It depends how one defines a t-distribution - as a ratio of a standard normal to a chi, or by its pdf or cdf. When I learned it, it was defined as the ratio, so a standard normal divided by a chi is a t-distribution by definition. One then has to do the work to derive the pdf and cdf from that definition.

Alternatively, if one defines a t-distribution by its pdf or cdf, one has to do the work to prove that a RV that is the ratio of a std normal to a chi has that pdf or cdf.

WWGD
andrewkirk said:
The distribution of a random variable that is the square root of a chi-squared distribution with k degrees of freedom is the chi distribution with k degrees of freedom. Details are here.
the claim is incorrect. It's not another chi-squared, it's a chi.
It depends how one defines a t-distribution - as a ratio of a standard normal to a chi, or by its pdf or cdf. When I learned it, it was defined as the ratio, so a standard normal divided by a chi is a t-distribution by definition. One then has to do the work to derive the pdf and cdf from that definition.

Alternatively, if one defines a t-distribution by its pdf or cdf, one has to do the work to prove that a RV that is the ratio of a std normal to a chi has that pdf or cdf.
Thanks, so a chi can coincide with a normal? I know there are issues with the square root not being an injection ( or not bijection once we choose one of the roots). I guess this is why the square root of a squared normal is not a normal? Or do we just consider the composition to be an absolute value?

I understand it comes down to a change of variable and using the Jacobian but I am writing on my phone since my PC is down.

WWGD said:
I know there are issues with the square root not being an injection ( or not bijection once we choose one of the roots). I guess this is why the square root of a squared normal is not a normal?
Yes, that's the reason. See the blue line in this graph, which is the chi distribution with one degree of freedom, which is the +ve sqrt of a squared std normal RV. It looks just like half a bell curve, and it is. If you insert ##k=1## in the formula given for the pdf you get double the pdf of a standard normal.

WWGD

## What is the distribution of the ratio between a standard normal and a square root chi-squared variable?

The distribution of the ratio between a standard normal and a square root chi-squared variable is known as the F-distribution. It is a continuous probability distribution that arises when comparing the variances of two independent normal distributions.

## What does the F-distribution tell us about the relationship between the two variables?

The F-distribution tells us about the ratio of variances between the two variables. It can be used to determine if there is a significant difference between the variances of the two populations.

## What is the formula for the F-distribution?

The formula for the F-distribution is F(x) = (x^(df1/2) * (df1 + df2)^(df1 + df2)/2) / ((df1/2) * (df2/2) * B(df1/2, df2/2)), where x is the value of the random variable, df1 and df2 are the degrees of freedom of the two variables, and B is the beta function.

## How is the F-distribution related to ANOVA?

The F-distribution is closely related to analysis of variance (ANOVA). ANOVA is a statistical test used to compare the means of three or more groups, and it uses the F-distribution to determine if there is a significant difference between the variances of these groups.

## What are some common applications of the F-distribution?

Some common applications of the F-distribution include ANOVA, regression analysis, and quality control. It is also used in various fields such as engineering, economics, and biology for hypothesis testing and comparison of variances.