response of LTI system to a Random Input Signal what is LTI filter? what is the output when x(t) is input?


let x(t) be the input signal to the system and y(t) denote the output signal. The output of the system may be expressed in terms of convolution integral as




Why we are not taking average of h(t) also? 


How comes E(X(t-tau)) to mx?
There are some basics that you really need to understand. The expectation operator of a deterministic (non-random) function just equals the function. Since the impulse response $h(t)$ is considered to be deterministic, you have

$$E[h(\tau)X(t-\tau)]=h(\tau)E[X(t-\tau)]$$

Furthermore, since $X(t)$ is known to be stationary, its mean doesn't depend on time, so you have

$$E[X(t-\tau)]=E[X(t)]=m_x$$