Last year a man wrote 142 checks. Let random variable x represent the number of...

Let X be the number of checks in 1 days
The rate of 142 checks in 365 days is same as 142/365 * 1=0.389041 in 1 days
X follows Poisson distribution with parameter (rate) ${\displaystyle \lambda =0.389041}$
X ${\displaystyle \sim }$ Poisson (${\displaystyle \lambda =0.389041}$)
Mean of poisson random variable is equal to its rate parameter ${\displaystyle \lambda }$
Mean or expected value, ${\displaystyle E\left(X\right)=\mu =0.3890}$
Variance of process is equal to its rate parameter ${\displaystyle \lambda }$
Variance, ${\displaystyle V\left(X\right)={\sigma }^2=0.3890}$
Standard deviation of poisson random variable is equal to square root of its rate parameter ${\displaystyle \lambda }$ i.e ${\displaystyle \sqrt{x}=\sqrt{0.389041}=0.623732}$
Standard deviation, ${\displaystyle \sigma =0.6237}$

The mean number of checks written per day is obtained as shown below:
Let x denotes the number of checks written per day.
From the information given, last year a person wrote 142 checks which means ${\displaystyle \lambda =0.3890(=\frac{142}{365})}$.
The mean is,
${\displaystyle \mu =\lambda }$
=0.389
The mean number of checks written per day is 0.389.
The variance is obtained as shown below:
The variance is,
${\displaystyle {\sigma }^2=\lambda }$
=0.389
The variance is 0.389.
The standard deviation is obtained as shown below:
The standard deviation is,
${\displaystyle \sigma =\sqrt{x}}$
${\displaystyle =\sqrt{0.389}}$
=0.624
The standard deviation is 0.624.
The mean number of checks written per day is 0.389.
The variance is 0.389.
The standard deviation is 0.624.