Understanding the exponential distribution I'm trying to wrap my head around the Exponential distribution and the meaning of its parameter. The parameter is the rate, right? So take, e.g.,
$$X\sim \exp(0.05)\,.$$
Now the probability of failure during the first time period is:
$$P(X\le1)=1-P(X>1)=1-e^{-0.05}=0.04877\,.$$
Now I can do the math and get the correct result and so on, but cannot wrap my head around why the result should be slightly less than 5%, rather than exactly 5%. I don't get the intuition behind it. 
 A: Suppose you have some mildly radioactive substance so that you expect to wait $20$ seconds between decays. That is a rate of $1/20$ per second. The average number of decays in one second is $1/20$. 
One way to get an average count of $1/20$ would be if the count were $1$ with probability $1/20$ and $0$ with probability $19/20$. However, sometimes there are $2$ or more decays in that second. For the average count to be $1/20$, when it is sometimes $2$ or more, the probability that you get a count of $0$ must be greater than $19/20$. Therefore, the probability that you wait more than one second before the first decay is greater than $19/20$, and the probability that the first decay occurs within the first second is less than $1/20$. 
A: There is a well known approximation for $e^x$ when $x$ is very close to zero:
$$ e^x \approx 1 + x $$
If we take $ x = -0.05 $, we get $ e^{-0.05} \approx 1 - 0.05 = 0.95 $.
So we can say:
$$ 1 - e^{-0.05} \approx 1 - 0.95 = 0.05 $$
You make a mistake when you think that a good approximation is the same as the exact value: the approximation is close to the exact value but rarely is the same number. The exact value is:
$$ 1 - e^{-0.05} = 1 - 0.951229 = 0.0487705 $$
The exact value of a calculus is difficult to reach. For an engineer or a scientific is better to use a good approximation.
