Question

Rate data often follow a lognormal distribution. Average power usage (dB per hour) for a particular company is studied and is known to have a lognormal distribution with parameters μ = 4 and σ = 2. What is the probability that the company uses more than 270 dB during any particular hour?

Answer 1

Answer: The probability is 0.0005%

Step-by-step explanation:

We are told that the data follows a lognormal distribution, hence we can use the formula for the probability given by:

$f_{X}(x)=\frac{1}{x} \cdot \frac{1}{\sigma \sqrt{2 \pi}} \exp \left(-\frac{(\ln x-\mu)^{2}}{2 \sigma^{2}}\right)$

Where, in this case, we have $\mu =4 and \sigma=2. To fin the probability we only need to evaluate the formula for x = 270 dB:[tex]f_{X}(270)=\frac{1}{270} \cdot \frac{1}{2 \times \sqrt{2 \pi}} \exp \left(-\frac{(\ln 270-4)^{2}}{2\times 2^{2}}\right)\\f_{X}(270)= 1.5\times10^{-5} = 0.00005 = 0.0005\%$

We fin out that the probability that the company uses more thant 270dB during any particular hour is very low, 0.0005%, for the given the parameters μ = 4 and σ = 2 in the lognormal distribution.