June 2018 p61 q4
3135
The random variable \(X\) has the distribution \(N(\mu, \sigma^2)\), where \(3\sigma = 4\mu\) and \(\mu \neq 0\). Find \(P(X < 3\mu)\).
Solution
We start by standardizing the variable. We want to find \(P(X < 3\mu)\).
Standardizing, we have:
\(P(X < 3\mu) = P\left( z < \frac{3\mu - \mu}{4\mu/3} \right)\)
\(= P\left( z < \frac{9\sigma/4 - 3\sigma/4}{\sigma} \right)\)
\(= P\left( z < \frac{6}{4} \right)\)
Using the standard normal distribution table, \(P(z < 1.5) = 0.933\).
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