Question $3$

$[6$ points$]$

A random variable $x$ is known to be Poisson distributed with $\frac{?}{\text{b}\text{a}\text{r}}(b)=4$ :

$f_x(x)=e^{-b}{\displaystyle \underset{k=0}{\overset{}{}}}\frac{b^k}{k!}(x-k)$

$($a$)$ Plot the density $f_x(x)$ and distribution $F_x(x)$ functions for $0x5$

$($b$)$ What is the probability of the event

Q$5$: State whether each of the following statements is True or False. Correct the false statements

A type $-1$ system track the ramp inputs with no steady state error.

Oscillation frequency is determined by the real part of the closed loop system poles.

The root locus starts at closed loop poles.

State variables are unique for a system

First order system response can be oscillatory.

Undamped system has its poles on the real axis

The two poles $5+j4$ and $10+j4$ have different peak time

Damped frequency is related to natural frequency by $w_d=w_n$

The system order and the system type are the same

In Rauth's table, the number of poles in the right half of s$-$plane is determined by the number of negative entries of the first column

The system transfer function depends on the system input

The poles $3+j4$ and $12+j16$ have the same overshoot