# Integrating factor question?

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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?

a. For what value of

b. Find

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

##### 1 Answer

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#

**normalisation constant** .

So:

And your PDF is:

The mean

#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: