The exercise asks for probabilities from a standard normal distribution (i.e., a Gaussian distribution with a mean of 0 and variance of 1) rounded to $10^{-4}$.

Let's calculate these probabilities using the properties of the standard normal distribution $X \sim N(0, 1)$:

1. $P(X \leq 1.85)$
2. $P(X \leq -1.18)$
3. $P(X > 2.58)$
4. $P(|X| \leq 1.96)$
5. $P(|X| \leq 2.8)$

Standard normal distribution properties:

For this, we use the cumulative distribution function (CDF) and symmetry properties:

• The CDF, $\Phi(x)$, gives $P(X \leq x)$.
• To calculate $P(X > x)$, we use: $P(X > x) = 1 - \Phi(x)$.
• For absolute values like $P(|X| \leq x)$, we use: $P(|X| \leq x) = 2\Phi(x) - 1$ due to the symmetry of the normal distribution.

I'll compute these probabilities.Here are the calculated probabilities, rounded to $10^{-4}$:

1. $P(X \leq 1.85) = 0.9678$
2. $P(X \leq -1.18) = 0.1190$
3. $P(X > 2.58) = 0.0049$
4. $P(|X| \leq 1.96) = 0.9500$
5. $P(|X| \leq 2.8) = 0.9949$

Let me know if you need more details or further explanation!

Here are some related questions:

1. How is the standard normal distribution derived from any normal distribution?
2. What is the difference between the CDF and the PDF of a normal distribution?
3. How can you use Z-scores to compare different datasets?
4. What is the significance of values like 1.96 in statistical hypothesis testing?
5. How can you use a standard normal table to approximate probabilities?

Tip: In a standard normal distribution, about 68% of values lie within 1 standard deviation, 95% within 2, and 99.7% within 3.