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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.

Soit X une gaussienne centrée réduite. Calculer à 10^{-4} près les probabilités suivantes : (1) P(X ≤ 1.85) (2) P(X ≤ -1.18) (3) P(X > 2.58) (4) P(|X| ≤ 1.96) (5) P(|X| ≤ 2.8).

Learn how to calculate probabilities for a standard normal (Gaussian) distribution. This example includes calculating probabilities like P(X ≤ 1.85) and P(X > 2.58), using key statistical concepts like the CDF of a normal distribution. Suitable for university-level probability and statistics.

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