Question

LetXhave the lognormal distribution with parameters 3 and 1.44. Find the probability thatX≤6.05.

Short answer

­\[P\left( {X \le 6.05} \right) = 0.1587\]

Step-by-step solution

Given information

X is a lognormal random variable.

\[\begin{array}{l}\mu = 3\\\sigma = \sqrt {1.44} \end{array}\]

Calculate the probability 

\[\begin{array}{c}X \le 6.05\\\log \left( X \right) \le \log \left( {6.05} \right) = 1.8\end{array}\]

\[\begin{array}{c}\Phi \left( {\frac{{X - \mu }}{\sigma }} \right) = \Phi \left( {\frac{{X - 3}}{{\sqrt {1.44} }}} \right)\\ = \Phi \left( { - 1} \right)\\ = 0.1587\end{array}\]

Hence,\[P\left( {X \le 6.05} \right) = 0.1587\].