I have to show that f(y) is a valid probability density function (pdf). I have integrated f(y) (I will attach the question & my working) but I am now stuck and not sure what to do next. Can someone please help me with this question?

Question

anonymous · Answer

attachment

blockcolder · Answer

The document is blurry. Please type it here. :(

anonymous · Answer

Does this look better? It looks OK when I open it. Not sure how to type all the maths as a post...

blockcolder · Answer

Loads better. :D
You know that you can split the real number line into the ff intervals:
$$\large (-\infty, -\theta), (-\theta, \theta), (\theta, +\infty)$$.
Split the integral accordingly and tell me what you get.

blockcolder · Answer

My bad. Combine the first 2 intervals.

anonymous · Answer

I know how to split an integral of the form $$\int\limits x^2/x^2+1  dx$$ which equals $$x^2+1/x^2+1 - 1/x^2+1 dx$$ but i am completely confused by$$\theta$$

anonymous · Answer

The intervals confuse me as well!

blockcolder · Answer

No, I meant split the integral like this:
$$\Large \int_{-\infty}^{\theta} f(y)\ \text{d}y + \int_{\theta}^{\infty}f(y)\ \text{d}y$$
Now, since for y<theta, f(y)=0, the first integral is 0.
All that's left is to prove that the second integral is equal to one by evaluating it.

anonymous · Answer

Great - many thanks for your help :)

blockcolder · Answer

No problem. :D