[Math] Finding probability given mean and standard deviation

probabilityself-learningstandard deviationstatistics

I don't know how to approach this problem:

X is normally distributed with a mean of 200 and a standard deviation of 10.

Find P(X ≥ 203)

Best Answer

Standardize so that you can use the standard normal distribution to find the numerical value for the answer. So, we have $$\begin{align} \mathbb{P}(X \geq 203) &= \mathbb{P}\left(\frac{X - 200}{10} \geq \frac{203 -200}{10}\right) \\ &= \mathbb{P}\left(Z \geq \frac{3}{10}\right) \\ &= 1 - \mathbb{P}\left(Z < \frac{3}{10}\right) \\ &= 1-\Phi\left(\frac{3}{10}\right) \,\,, \end{align}$$ where $Z \sim N(0,1)$ and $\Phi$ is the standard normal cumulative distribution function.

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