We're given a random variable $X$ and it's distribution function as $F_X (x)$. Also , $F_x (x)$ can be written as a function of $( x - \alpha )/ \beta$ , that is $x, \alpha , \beta$ appear in $F_X(x)$ only in the indicated form.
We need to show that if $\alpha$ is increased by $ \Delta\alpha$ , then so does the mean of $X$.
What I tried :
We're given $F_X (x) = h(y)$ where $y = ( x - \alpha )/ \beta$. { $h$ is any arbitrary function }
Also , $E(X) = \int_0^{\infty}(1 - F_X(x) )dx - \int_{- \infty}^0F_X(x)dx$
=> $E(X) = \int_0^{\infty}(1 - h(y) )dx - \int_{- \infty}^0h(y)dx$
=> $E(X) = \int_0^{\infty} \beta(1 - h(y) )dy - \int_{- \infty}^0\beta h(y)dy$
I am not able to proceed further , can anyone help ?
This is clearly about the scale-location family.
Define another random variable $$Y \equiv \frac{X- \alpha}{\beta} \quad, \quad \text{or} \quad X = \alpha + \beta Y$$
Then simply $~E(X) = E(\alpha + \beta Y) = \alpha + \beta E(Y)~$ leading to $\alpha \to \alpha + \Delta\alpha \implies E(X) = \alpha +\Delta\alpha + \beta E(Y)$