Math Problem Statement

Given the normally distributed random variable X ~ N(μ = 35, σ = 10), find P(X < 40).

Solution

The problem in the image reads:

"Given the normally distributed random variable XN(μ=35,σ=10)X \sim N(\mu = 35, \sigma = 10), find P(X<40)P(X < 40)."

Solution:

The random variable XX follows a normal distribution with mean μ=35\mu = 35 and standard deviation σ=10\sigma = 10. We need to find the probability P(X<40)P(X < 40).

First, we standardize the variable XX using the Z-score formula:

Z=XμσZ = \frac{X - \mu}{\sigma}

For X=40X = 40:

Z=403510=510=0.5Z = \frac{40 - 35}{10} = \frac{5}{10} = 0.5

Now, we need to find the probability P(Z<0.5)P(Z < 0.5).

Using the standard normal distribution table or a calculator, the probability for Z<0.5Z < 0.5 is approximately:

P(Z<0.5)0.6915P(Z < 0.5) \approx 0.6915

Final Answer:

P(X<40)0.6915P(X < 40) \approx 0.6915

Would you like further details or have any questions?

Here are some related questions:

  1. What is the probability P(X>40)P(X > 40) for the same distribution?
  2. How do you calculate the Z-score for a value less than the mean?
  3. What is P(X<35)P(X < 35) given XN(μ=35,σ=10)X \sim N(\mu = 35, \sigma = 10)?
  4. How does changing the mean μ\mu affect the probability P(X<40)P(X < 40)?
  5. Can you find P(30<X<40)P(30 < X < 40) for the given distribution?

Tip: When working with normal distributions, always convert to the Z-score first, as it allows you to use standard tables or tools for probability calculations.

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Math Problem Analysis

Mathematical Concepts

Normal Distribution
Probability

Formulas

Z-score formula

Theorems

Properties of Normal Distribution

Suitable Grade Level

Advanced High School / College