The task involves a normally distributed random variable $X \sim N(100, 100)$, and we are solving three parts. Let’s break this problem down:

a) $\mathbb{P}(X \leq 100)$

To calculate this probability:

1. The random variable $X$ has a mean $\mu = 100$ and a variance $\sigma^2 = 100$, so the standard deviation $\sigma = \sqrt{100} = 10$.
2. Standardize $X$ to $Z$, where $Z \sim N(0, 1)$: $Z = \frac{X - \mu}{\sigma}$ For $X = 100$: $Z = \frac{100 - 100}{10} = 0$
3. From the standard normal table, $\mathbb{P}(Z \leq 0) = 0.5$.

Thus, $\mathbb{P}(X \leq 100) = 0.50$.

b) Standardized normal distribution value for the above probability

The probability $\mathbb{P}(X \leq 100) = \mathbb{P}(Z \leq z)$ corresponds to $z = 0$, because we standardized $X$ in part (a).

So, $z = 0$.

c) Find $z$ where $\mathbb{P}(Z \leq z) = \frac{17}{20}$

To calculate this:

1. $\mathbb{P}(Z \leq z) = \frac{17}{20} = 0.85$.
2. Using the standard normal table or inverse normal function, the $z$-score corresponding to $0.85$ is approximately: $z = 1.04$

Final Answers:

1. a) $\mathbb{P}(X \leq 100) = 0.50$
2. b) $z = 0$
3. c) $z = 1.04$

Would you like a detailed explanation for any part?

Here are five related questions for deeper understanding:

1. How do you calculate the standard score $Z$ for any given $X$?
2. What does the standard normal distribution represent?
3. Why does $\mathbb{P}(X \leq \mu)$ equal 0.5 in a symmetric normal distribution?
4. How can you calculate probabilities without standard normal tables?
5. What are common applications of normal distributions in real life?

Tip: Always verify whether probabilities correspond to areas under the curve when interpreting normal distributions.