PDF highest density interval sketches

[SirEllwood]

1. Sketch a probability density function for which the highest density interval (HDI) takes the form:

(i) (a, b)
(ii) (a, b) U (c, d)
(iii) (a, b) U (c, d) U (e, f).
where a < b < c < d < e < f.

3. Now i believe that the graphs are going to look something like like e^{-x^2}, x^2*e^{-x^2), (x^2-1)^2*e^{-x^2} for the 3 questions respectively? But I am not sure how i am going to label my graphs with regard to the intercepts, and maximum and minimum points etc.

Anybody shed some light on this?

 

[lippyka]

1. (i) The probability density function for this HDI would take the form of a bell-shaped curve with its maximum point at x = (a+b)/2, and an inflection point at x = (b+c)/2. The curve would be symmetric about the line x = (a+b)/2. 2. (ii) The probability density function for this HDI would take the form of a double bell-shaped curve with two maximum points at x = (a+b)/2 and x = (c+d)/2 respectively. The curve would be symmetric about the lines x = (a+b)/2 and x = (c+d)/2. 3. (iii) The probability density function for this HDI would take the form of a triple bell-shaped curve with three maximum points at x = (a+b)/2, x = (c+d)/2 and x = (e+f)/2 respectively. The curve would be symmetric about the lines x = (a+b)/2, x = (c+d)/2, and x = (e+f)/2.