L3 examples

Problem 1

In going through some historical records, you find that scientists from a lost civilization tried to measure the distance from the ground to some clouds. Based on the data you assume that the distance is a Gaussian random variable with a mean of 1830m and a standard deviation of 460m. What is the probability that the clouds would be at a height above 2750m?

Solution

Let \(X\) be this Gaussian random variable. This problem essentially requires us to work out \(p \left( X > 2750 \right)\). This can be expressed as

\[ \large p \left(X > 2750 \right) = 1 - p \left( X \leq 2750 \right) = 1 - \Phi \left( z \right) \]

where \(z = (2750 - 1830)/460 = 2\). Thus we have

\[ \large 1 - \Phi \left( 2 \right) = 1 - 0.9772 = 0.0228 \]

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## Problem 1

In going through some historical records, you find that scientists from a lost civilization tried to measure the distance from the ground to some clouds. Based on the data you assume that the distance is a Gaussian random variable with a mean of 1830m and a standard deviation of 460m. What is the probability that the clouds would be at a height above 2750m?

<details>
<summary>Solution</summary> 
    
Let $X$ be this Gaussian random variable. This problem essentially requires us to work out $p \left( X > 2750 \right)$. This can be expressed as
    
$$
\large
p \left(X > 2750 \right) = 1 - p \left( X \leq 2750 \right) = 1 - \Phi \left( z \right) 
$$
    
where $z = (2750 - 1830)/460 = 2$. Thus we have 
    
$$
\large
1 - \Phi \left( 2 \right) = 1 - 0.9772 = 0.0228
$$
</details>