(Solved): Can you please show me your work please? I already answered number 1. Can you please answer number 2 ...

Can you please show me your work please? I already answered number 1. Can you please answer number 2. Thank you! 

4. The random variable \( \mathrm{X} \) is the number of the number of calls received by a switchboard. Suppose \( \mathrm{X} \) follows a Poisson distribution and the average number of occurrences in twenty minutes is two. (1) What is the probability that between \( 10: 00 \) and 10:30 the switchboard will receive exactly five calls? \[ \begin{array}{l} \mathrm{P}(\mathrm{x}=\mathrm{x})=\lambda^{\wedge} \mathrm{x}^{*} \mathrm{c}^{\wedge}(-\lambda) / \mathrm{x} ! \\ \mathrm{P}(\mathrm{x}=5)=2^{\wedge} 4^{*} \mathrm{e}^{\wedge}-2 / 5 ! \\ \mathrm{P}(\mathrm{x}=5)=0.0180 \end{array} \] (3) What is the probability that between 11:00 and 11:30 the switchboard will receive more than two calls but fewer than six calls?