Two normal distributions in the form are shown.
The blue curve is and the black curve is with parameters controlled by the input boxes.
The shaded areas represent the probabilities shown, you can move the points to change the probabilities.

Tasks

Set the black curve to and find .

Find the value of such that .

Set the black curve to and find .

Find the value of such that .

What do you notice?

Set the black curve to and find .

Find the value of such that .

Set the black curve to and find .

Find the value of such that .

What do you notice?

Test your findings by trying to find if using the red area only, then check your answer.