Lesson 76 — Normal distribution

$X \sim \mathcal N(70, 10^2)$. Calculate the z-score for $X = 85$.

$X \sim \mathcal N(100, 15^2)$. Calculate the z-score for $X = 80$.

Calculate $P(Z \leq 1{,}96)$.

Calculate $P(Z \geq 1{,}96)$.

Calculate $P(-1{,}96 \leq Z \leq 1{,}96)$.

Calculate $P(Z \leq -1{,}5)$.

Calculate $P(0 \leq Z \leq 2)$.

$X \sim \mathcal N(50, 10^2)$. Calculate $P(X > 65)$.

$X \sim \mathcal N(50, 10^2)$. Calculate $P(40 < X < 60)$.

$X \sim \mathcal N(0, 4)$ (variance = 4). Calculate $P(X > 3)$.

Calculate the 90th percentile of $\mathcal N(100, 15^2)$.

Calculate $Q_1$ (25th percentile) of $\mathcal N(100, 15^2)$.

IQ $\sim \mathcal N(100, 15^2)$. What % of the population has an IQ between 85 and 115?

IQ $\sim \mathcal N(100, 15^2)$. What % has an IQ above 130?

IQ $\sim \mathcal N(100, 15^2)$. What % has an IQ above 145?

Heights $\sim \mathcal N(170, 8^2)$ cm. What % has a height above 186 cm?

Grades $\sim \mathcal N(70, 10^2)$. From which grade does the top 5% start?

$X \sim \mathcal N(0, 1)$. Calculate $P(|X| > 3)$.

For $X \sim \mathcal N(\mu, \sigma^2)$, what is the relationship between median, mode, and $\mu$?

$X \sim \mathcal N(20, 4^2)$ and $Y \sim \mathcal N(10, 3^2)$ independent. What is the distribution of $X + Y$?

Monthly salary $\sim \mathcal N(5000, 1500^2)$ reais. What is the salary floor for the top 10%?

Flight duration $\sim \mathcal N(120, 10^2)$ min. How much time to reserve to have 99% confidence of arriving on time?

Daily stock returns $\sim \mathcal N(0{,}001,\; 0{,}01^2)$. Calculate $P(\text{loss} > 2\%)$.

Voltage $\sim \mathcal N(220, 5^2)$. Device fails if $V > 235$ V. Calculate the probability of failure.

Parts with diameter $\mathcal N(10{,}00;\; 0{,}02^2)$ mm. Tolerance $10{,}00 \pm 0{,}05$ mm. What fraction is rejected?

Survey with 1000 respondents estimates real proportion $p = 0{,}50$. Construct a 95% CI for $p$.

$\bar X = 105$, $n = 25$, $\sigma = 10$ (known). Construct a 95% CI for $\mu$.

Test $H_0: \mu = 100$ vs. $H_1: \mu \neq 100$. $\bar X = 105$, $n = 25$, $\sigma = 10$. Calculate the p-value and decide.

Execution time $\sim \mathcal N(50, 5^2)$ ms. To guarantee SLA with 95% of requests below the limit, what threshold should be set?

Six Sigma: specification $\mu \pm 6\sigma$. With a 1.5σ adjustment for process drift, calculate defects per million. Why is the result 3.4 ppm and not virtually zero?

X-bar chart with $n = 5$, $\sigma = 2$ (known), $\bar{\bar{X}} = 100$. Calculate UCL and LCL at $\pm 3\sigma$.

ML model scores $\sim \mathcal N(0{,}80,\; 0{,}05^2)$. What is the threshold to select the top 20% of models?

100Ω resistor with ± 5% tolerance. Assuming $\sigma = 5/3$ ohm (3$\sigma$ = tolerance), calculate the fraction within specification.

Annual portfolio return $\sim \mathcal N(5\%, 20\%^2)$. Calculate the probability of a negative return in one year.

ENEM grades (Math) $\sim \mathcal N(520, 110^2)$. Calculate $P(\text{grade} > 700)$.

Annual IPCA modeled as $\mathcal N(4{,}5\%,\; 1{,}5\%^2)$. Inflation target: up to 6.5%. What is the probability of exceeding the target?

Why do we standardize to the standard normal? What justifies the existence of a single $\Phi(z)$ table?

Is the normal tail "thin" or "heavy"? Why does this matter in financial risk modeling?

Show that if $X \sim \mathcal N(0, 1)$, then $Y = X^2$ has a chi-squared distribution with 1 degree of freedom.

Demonstrate that if $X \sim \mathcal N(\mu, \sigma^2)$ and $Y = aX + b$ (with $a > 0$), then $Y \sim \mathcal N(a\mu + b,\; a^2\sigma^2)$.

Demonstrate that $\int_{-\infty}^{+\infty} e^{-x^2/2}\,dx = \sqrt{2\pi}$ using the polar coordinates trick.

Demonstrate (sketch) that the normal distribution maximizes differential entropy among all continuous distributions with fixed mean $\mu$ and variance $\sigma^2$.