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Michael Hardy
Michael Hardy
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A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Here is the actual question:

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Here is the actual question:

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?

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Michael Hardy
Michael Hardy
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  • 126

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Here is the actual question:

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Here is the actual question:

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?

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Michael Hardy
Michael Hardy
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Symmetry in the triangular distribution

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is $$ \frac{a+b+c} 3 $$ and the variance is $$ \frac{a^2+b^2+c^2-ab-ac-bc} {18}. $$ Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?