I was trying to solve this problem:
$f(x) = k[\phi (x) + \lambda g(X)]$
Where:
$\phi(x)$ is a normal distribution with mean =0 and variance =1.
$g(x)$ is defined as:
$\frac{1}{\lambda}$ for $ \leq x \leq \lambda $
0, otherwise.
SO, my question is, if we want to find the value of k, I would take the C.D.F of $f(x)$ to find the value of k in terms of $\lambda$.
When we are taking C.D.F, there are two options available:
$F(X)=k\int_{-\infty}^{\infty} \phi(x) dx + k\lambda \int_{-\infty}^{\infty} g(x)$ or:
$F(X)=k\int_{-\infty}^{\infty} \phi(x) dx + k\lambda \int_{0}^{\lambda} g(x)$.
So, I looked at the mark scheme, it chose the second integrals to get the F(x), my question is, why is that so? I dun quite understand why we can put different limits on it, its like both functions are having different inputs.
Thank you very much for you guy's replies.
We need $\int_{-\infty}^\infty f = 1$ as it should be a density function. We get $$ 1 = \int_{-\infty}^\infty f = k\bigg[\int_{-\infty}^\infty \phi + \lambda \int_{-\infty}^\infty g \bigg] = k[1+ \lambda]$$ cause both $\phi$ and $g$ are density functions. Then $$k = \frac {1}{\lambda+1}$$