Thread: Exponential distribution with an unknown parameter, theta

1. Exponential distribution with an unknown parameter, theta

I don't actually know what topic this falls under exactly to be honest. The question goes:

In an experiment to study pulses along a nerve fibre, the times between 101 successive pulses were measured. The $n=100$ observations gave a sample mean $\bar{x}=26.05$ seconds. Assuming that these data are a random sample from an exponential $Exp(\theta)$ distribution, calculate

(i) the most likely value for the mean pulse rate $\theta$ per second

(ii) the most likely value for the probability $\phi$ that the time between successive pulses is greater than 20 seconds i.e. $\phi=P(X>20)=e^{-20\theta}$.

2. Originally Posted by chella182
I don't actually know what topic this falls under exactly to be honest. The question goes:

In an experiment to study pulses along a nerve fibre, the times between 101 successive pulses were measured. The $n=100$ observations gave a sample mean $\bar{x}=26.05$ seconds. Assuming that these data are a random sample from an exponential $Exp(\theta)$ distribution, calculate

(i) the most likely value for the mean pulse rate $\theta$ per second

(ii) the most likely value for the probability $\phi$ that the time between successive pulses is greater than 20 seconds i.e. $\phi=P(X>20)=e^{-20\theta}$.
Use the sample mean as an estimator of the mean for the exponential distribution.

$\Pr(\theta > 20) = \int_{20}^{+\infty} f(\theta) \, d \theta$ where $f(\theta)$ is the pdf.

3. I don't remember learning this at all nothing like this is in my notes.

Why integrate between 20 and $\infty$?