UMVE_one_sample_poisson_cut_distribution.qmd

---
title: "UMVUE one sample cut poisson distributin"
format: 
    html:
        code-fold: true
jupyter: python3
---
$$
\begin{align*}
& \text{if} ~~f_X(x) = \frac{\exp(-\theta)\times \theta^x}{(1 - \exp(-\theta)) \times x!}, \quad x =  1, 2, 3, \dots\quad \theta > 0,  \\
& \gamma(\theta) = 1 - \exp(-), \\
& \text{1. calculate UVMUE for} ~\gamma(\theta) ~\text{and Variance of this estimator}.\\ 
& \text{2. calculate Crammer-Rao Bound for above estimator}.
\end{align*}
$$

```{python}
from sympy.stats import DiscreteRV, P, E, variance
from sympy import Symbol, factorial, exp, S, simplify, init_printing
x = Symbol('x', integer = True, positive = True)
t = Symbol('theta', real = True, positive = True)
pdf = exp(-t) * t**(x) /(factorial(x) * (1-exp(-t)))
D = DiscreteRV(x, pdf, set = S.Naturals)

# get expectation of X
E(D)


```


# get variance of X
```{python}
variance(D)

```


## get probability of P(X >= 1)
```{python}
temp1 = P(D >= 1)
simplify(temp1)
```




```{python}
temp1 = P(D >= 2)
simplify(temp1)


```


```{python}
temp1 = P(D >= 3)
simplify(temp1)


```