Let's solve the problems one by one.

Problem 1:

Given:

• The mean ($\mu$) = 4.35 mrem
• The standard deviation ($\sigma$) = 0.59 mrem
• The amount of cosmic radiation is modeled as a normally distributed random variable.

We need to find the probabilities for the following cases:

(a) The probability that the radiation is between 4.00 and 5.00 mrem.

To find this probability, we convert the raw values to standard normal values (z-scores) using the formula: $z = \frac{x - \mu}{\sigma}$

1. For $x = 4.00$: $z_1 = \frac{4.00 - 4.35}{0.59} \approx -0.59$

2. For $x = 5.00$: $z_2 = \frac{5.00 - 4.35}{0.59} \approx 1.10$

We then find the probabilities corresponding to $z_1$ and $z_2$ using standard normal distribution tables or a calculator:

• $P(z < -0.59) \approx 0.2776$
• $P(z < 1.10) \approx 0.8643$

The probability that $x$ is between 4.00 and 5.00 mrem is: $P(4.00 < x < 5.00) = P(z < 1.10) - P(z < -0.59) \approx 0.8643 - 0.2776 = 0.5867$

(b) The probability that the radiation is at least 5.50 mrem.

First, find the z-score for $x = 5.50$: $z_3 = \frac{5.50 - 4.35}{0.59} \approx 1.95$

The probability that $z$ is at least 1.95 is: $P(z > 1.95) = 1 - P(z < 1.95) \approx 1 - 0.9744 = 0.0256$

Problem 2:

Given:

• The probability density function is $f(x) = k e^{-x/3}$ for $x > 0$, and $f(x) = 0$ for $x \leq 0$.

We need to find the value of $k$.

To satisfy the properties of a probability density function, the integral of $f(x)$ over the entire range must equal 1: $\int_{0}^{\infty} k e^{-x/3} \, dx = 1$

Calculating the integral: $k \int_{0}^{\infty} e^{-x/3} \, dx = 1$ The integral of $e^{-x/3}$ is $-3 e^{-x/3}$, so: $k \left[ -3 e^{-x/3} \right]_{0}^{\infty} = 1$ $k \left[ 0 - (-3) \right] = 1$ $3k = 1$ Thus, $k = \frac{1}{3}$.

Would you like more details, or do you have any further questions?

Related Questions:

1. What is a z-score and how is it used in probability calculations?
2. How do you find probabilities using the standard normal distribution table?
3. What is the integral of a function and why is it used in probability density functions?
4. How can the properties of a normal distribution be applied to solve real-world problems?
5. What is the significance of standardizing a normal distribution?

Tip:

When dealing with probabilities for normal distributions, converting raw scores to z-scores simplifies the calculations by using standard normal distribution tables.