Question

Consider the random process

X(t) = U + Vt.

where U is a zero-mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are statistically independent. The mean value of the random process at t = 2 is

• A. 2

Answer 1

The correct option is A 2

Given random process
x(t) = u + vt
Mean of random process
E[x(t)] = E[u + vt]

E[x(t)] = E[u] + E[vt] $(\therefore$ u and v are statistically independent)

E[x(t)] = 0 + E[v]t
$(\because$ E[u] is given as zero

E[x(t)] = E[v]t

Since, v is a random variable uniformly distributed between 0 and 2.

where $f\left(v\right)=(\begin{array}{cc}\frac{1}{2-0},& 0<v<2\\ 0,& \mathrm{otherwise}\end{array}$is a p.d.f. for Uniform Random Variable v.

$E\left[v\right]={\int }_{-\infty }^{\infty }{f}_v\left(v\right)dv={\int }_0^2\left(\frac{1}{2}\right)dv=\frac{1}{2}[v{]}_0^2=1$

At t = 2, E[x(t)] = E[v] × 2

= 1 × 2 = 2