Confused how to do this probability question

Question

The hourly median power (in decibels) of received radio signals transmitted between two cities follows a lognormal distribution with parameter values u = 3.5 and o(sigma)=1.2

a. What is the 90th percentile of this distribution?

b. What is the probability that received power for one of these radio signals is

less than 150 decibels?

c. Consider a random sample of 10 radio signals. What is the probability that for 6 of these signals, the received power is less than 150 decibels?

Answer 1

Hi Bob J.,

Frequently lognormal distributions are used to analyse engineering failure situations. However, I guess inadequate transmission strength might count as that! There are formidable-looking equations out there, because those conversions sure do spawn parameters!

I'm not entirely sure, but I think the µ and σ apply immediately to the lognormally-graphed x-axis values for hourly median power. They would look weird plotted on non-log-transformed power data -- the error bar for σ would go way to the right from the mean. The graph is (y) vs. (x) is (probability of occurrence) vs. (log(hourly median power)). So for your queries, you would just treat the statistics as applying to that Gaussian-appearing graph. But, once you have determined the limit or limits values (on the x-axis), remember to convert them back into regular units by taking their 10^x values.

So for (a), you would read a standard probability table to find the 5%-excluded limits on either side, then scale those (unless the table showed for multiple σ values, don't think that's customary though) for the existing σ value. Take those two points (which are in log(power) form) and upscale them to power, and there you are.

For (b), you would first find log(150), take the difference of that to μ in units of σ, and read the table for the probability (essentially, the reverse of what you did in (a)?). Note that all the statistics take place using the lognormal data, but answers for power are in un-logged-normal converted form!

Reckon you can handle the rest?

-- Cheers, __Mr d.