Question

Integrating factor question?

a. For what value of `\lambda` is the following ( `\int_1^2 \lamda\root(3)(x^4)dx` ) a density function?
b. Find `\mu`

Also: does anyone know where I can find practice questions on this type of problem?

Answer 1

See below

Explanation:

What you have here is a "wannabe" probability density function (PDF):

• `mathbf P (x) = \lambda \ root(3)(x^4)`, for `1 le x le 2`.

But first you want to normalise the PDF, ie make this become true:

• `int_1^2 mathbf P(x) = 1`.

....which is the continuous analogue of the requirement of a discrete distribution:

• `sum_i P(x_i) = 1`

`lambda` is your normalisation constant .

So:

`\ int_1^2 \lambda \ root(3)(x^4)\ dx =lambda \ ( (3 x^(7/3))/7 )_1^2`

`=lambda * 3/7(4 root(3) 2 - 1) color(red)(= 1)`

`implies lambda = 7/ ( 3(4 root(3) 2 - 1)) approx 0.58 approx 7/12`

And your PDF is:

`mathbb P (x) = 7/ ( 3(4 root(3) 2 - 1)) \root(3)(x^4)`

The mean `mu` is given by:

• `mu = int_1^2 x \ mathbb P(x) \ dx`

....which is the continuous analogue of the mean of a discrete distribution:

• `mu = sum_i P(x_i) \ x_i`

Here:

`mu = lambda \ int_1^2 \ x^(7/3) \ dx`

`= lambda ( (3 x^(10/3))/10)_1^2`

`7/ ( 3(4 root(3) 2 - 1)) * 3/10 (8 root(3)2 - 1)`

`implies mu = (7 (8 root(3)2 - 1))/(10 (4 root(3)2 - 1)) approx 1.57`