# Find the p value following the exponential distribution $\mu=3$

I want to find the $$p$$-value (manually) of the following Hypothesis testing.

$$H_0:\mu\leq 3 \quad \text{vs} \quad H_1:\mu >3$$

The main thing I know is that

$$P(\mathrm{Re}\,j \mid \mu \leq 3)=P(X\geq 3 \mid \mu \leq 3)= e^{-1} \approx0.36$$

Can I use the $$z$$ value and use the formula probability of $$z$$? Or from where can I start?

Your null hypothesis is that your exponential distribution has a rate $$\mu$$ which is $$\leq 3$$. Your alternate hypothesis is that $$\mu \geq 3$$. Now, you get some observation, $$x$$. What is the probability that this sample is consistent with the null-hypothesis? Meaning, what is the probability that the null hypothesis would generate a sample $$\geq x$$? Conditional on $$\mu$$, this is simply $$e^{-\mu x}$$. Since your null hypothesis is that $$\mu \leq 3$$, you integrate over it to get the p-value:
$$p = \int\limits_0^3 e^{-\mu x}d \mu = \frac{1-e^{-3x}}{x}$$
• xAlso, it shouldn't be divided by $x$ instead of 3? – Lexie Walker Mar 13 at 17:30
• Yes, sorry.. fixed the typo. Yes, you need to have a very large $x$ for your p-value to be small. The larger the $x$, the smaller the chance an exponential with rate $<3$ generated it. – Rohit Pandey Mar 13 at 17:53