Can someone tell me how to read this equation in words? $$
P\left[\frac{\text{MTBF}\cdot2r}{\chi^2_{1-\alpha/2,2(r+1)}}
\le
\text{True MTBF}
\le
\frac{\text{MTBF}\cdot2r}{\chi^2_{\alpha/2,2r}}
\right]
\ge 1-\alpha
$$
Can someone please help me to understand this formula. I know/understand the variables but not its mathematical significance. 
 A: What you have is a probabilistic statement about the parameter sought (True MTBF) that says that the probability that the true parameter is between the two fractions is greater than or equal to 1-$\alpha$. This probably sound like a confidence interval, because it is. 
For a given $r$ and a given $\alpha$, the stuff in brackets defines a set (a confidence set). Say that $\alpha=0.05$, as is typical, and maybe $r=50$: then you can find the corresponding points on the chi-square distributions
$$\chi^2_{0.975, 102} = 131.84 \ \ \ \text{and} \ \ \  \chi^2_{0.025, 100}=74.22$$ 
Notice that the first $\chi^2$ critical point is larger and the second is smaller; as they're in the denominators and the numerators are constant, this ensures that the confidence set is of the form $[a,b]$ with $a<b$ which is intuitive. 
Now, let's say that the observed MTBF is $\hat{MTBF} = 5$ just for the sake of calculation. Then, 
$$\text{Pr}\left[ \frac{5*2*50}{131.84}\leq MTBF \leq \frac{5*2*50}{74.22} \right] \geq 0.95 \iff $$ $$\text{Pr}(3.79\leq MTBF \leq 6.74) \geq 0.95 \iff$$ $$\text{95% CI is}: (3.79,6.74)$$
A: This is how to read this equation in words, somewhat simplified:


*

*"The probability of the logical statement in the square brackets being true is greater than or equal to some high probability" (1.0 is the maximum probability, so this is only smaller than 1.0 by $\alpha$).

*"In the square bracket, there is a logical statement that is either true or false: the True Mean Time Between Failures (MTBF) is somewhere between a lower value and a upper value."

*"The formulas for 'lower value' and 'upper value' are confidence intervals:  the estimated MTBF (from your sample data) TIMES two times the number of failures $r$, DIVIDED by the Chi-squared value for the lower and upper limits, respectively, based on your confidence level $\alpha$ and $r$."


See @ScouserInTrousers answer for explanation of Chi-squared value, plus an example calculation.
