Let   X space tilde thin space N left parenthesis 0 comma 2 squared right parenthesis comma spacewhat is P(X<3)?

Given that \( X \sim N(0, 2^2) \), this means that \( X \) follows a normal distribution with a mean \( \mu = 0 \) and a standard deviation \( \sigma = 2 \). The probability we are interested in is \( P(X < 3) \).

To find this probability, we standardize the normal variable \( X \) to convert it to the standard normal distribution \( Z \), which has a mean of 0 and a standard deviation of 1. The standardization formula is:

\[
Z = \frac{X - \mu}{\sigma}
\]

Substituting the given values:

\[
Z = \frac{3 - 0}{2} = \frac{3}{2} = 1.5
\]

Now, we want to find \( P(Z < 1.5) \).

Using standard normal distribution tables or a calculator:

\[
P(Z < 1.5) \approx 0.9332
\]

Thus, \( P(X < 3) \approx 0.9332 \).

Would you like more details on this calculation or have any other questions?

Here are 5 related questions for further exploration:
1. How do you find the probability for \( P(X > 3) \) with the same distribution?
2. What is the probability that \( X \) lies between \(-1\) and \( 3 \)?
3. How does the probability change if the standard deviation is increased to 3?
4. What is the 95th percentile for the given distribution \( X \)?
5. How would you find the probability \( P(X < 3) \) using a normal distribution calculator?

**Tip:** Understanding how to standardize a normal distribution to the standard normal form is crucial for solving most probability questions involving the normal distribution.

Understand how to calculate the probability P(X < 3) for a normal distribution N(0, 4) with a mean of 0 and standard deviation of 2. Step-by-step solution using standardization and the standard normal distribution table.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation