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?

1 Answer
Apr 25, 2018

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#