# Help understanding bayesian update for exponential distribution

I'm following these slides. I have some lamps, which I expect to die at a time $$t$$ where $$p(t) = \lambda e ^{-\lambda t}$$ for some $$\lambda$$. My prior for $$\lambda$$ is given by $$p(\lambda) = \Gamma(\lambda, 2, 2)$$. I observe one lamp die at time 10. According to those slides, my posterior is $$k' = 2 + 1 = 3$$ $$\theta' = 2 / (1 + 2 * 10) = 0.095$$ so

$$p(\lambda) =\Gamma(\lambda, 3, 0.095)$$ When I plot this though, it doesn't look right: My posterior has shifted far to the left, even though my observation was to the right of the expectation of my prior. What's going on?

Code:

import matplotlib.pyplot as plt
import numpy as np
from scipy.special import gamma as gamma_function

def gamma(x, k, theta):
return np.power(x, k-1) * np.exp(-x/theta) / (gamma_function(k) * np.power(theta, k))

k = 2
theta = 2

data = 
new_k = k + len(data)
new_theta = theta / (1 + theta * sum(data))

x = np.linspace(0, 10, 100)
plt.plot(x, gamma(x, 2, 2), label = 'prior')
plt.plot(x, gamma(x, new_k, new_theta), 'r', label = 'posterior')
plt.legend()


The mean of an exponential RV is $$1/\lambda$$, so an increasing lifetime in data decreases your posterior $$\lambda$$, e.g. mean life=10 means $$\lambda=0.1$$. So, it's normal that your posterior belief goes towards left side of the plot.