For a normally distributed random variable $X$ with mean $\mu = 0$ and standard deviation $\sigma = 1$, the probability that $X$ falls within certain intervals can be related to the properties of the standard normal distribution (denoted as $Z \sim N(0, 1)$).

1. Probability that $X$ falls within the interval $[-1, 1]$:

We need to compute the probability $P(-1 \leq X \leq 1)$, which is the area under the standard normal curve from -1 to 1:

$P(-1 \leq X \leq 1) = P(Z \leq 1) - P(Z \leq -1)$

Using the cumulative distribution function (CDF) for a standard normal distribution, we have:

$P(Z \leq 1) \approx 0.8413 \quad \text{and} \quad P(Z \leq -1) \approx 0.1587$

So:

$P(-1 \leq Z \leq 1) = 0.8413 - 0.1587 = 0.6826$

This means approximately 68.26% of the values of a standard normal variable fall within one standard deviation of the mean (between -1 and 1).

2. Probability that $X$ falls within the interval $[0, 1]$:

Now, we compute $P(0 \leq X \leq 1)$, which is the area under the standard normal curve from 0 to 1:

$P(0 \leq Z \leq 1) = P(Z \leq 1) - P(Z \leq 0)$

Since $P(Z \leq 1) \approx 0.8413$ and $P(Z \leq 0) = 0.5$, we have:

$P(0 \leq Z \leq 1) = 0.8413 - 0.5 = 0.3413$

Thus, approximately 34.13% of the values of a standard normal variable fall between 0 and 1.

3. Relationship Between the Two Probabilities:

The probability of falling within $[-1, 1]$ is 68.26%, while the probability of falling within $[0, 1]$ is 34.13%. Since the standard normal distribution is symmetric around zero, the probability that $Z$ falls between -1 and 0 is also 34.13%, meaning that:

$P(-1 \leq Z \leq 1) = 2 \times P(0 \leq Z \leq 1)$

In other words, the probability of $Z$ falling within $[-1, 1]$ is exactly twice the probability of $Z$ falling within $[0, 1]$.

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Would you like any further details or explanations?

Here are five related questions to explore:

1. What is the probability of a standard normal variable falling between -2 and 2?
2. How does the probability change if the interval is expanded to $[-2, 2]$?
3. Can you derive the probability for intervals such as $[1, 2]$ using the CDF?
4. What are the properties of the normal distribution that lead to symmetry?
5. How does the concept of z-scores relate to probabilities in normal distribution?

Tip: Always remember that about 68.26%, 95.44%, and 99.73% of the data falls within 1, 2, and 3 standard deviations of the mean, respectively, in a standard normal distribution.