[A-L] (2023/24) Foglio 10 - Esercizio 2

[Elia-Belli]

Tavola integrale gaussiano

[Elia-Belli]

$X\sim \mathcal{W}(2,25)$ che riconduco ad una gaussiana standard $Z=\frac{X-\mu}{\sigma}=\frac{X-2}{5}\sim \mathcal{W}(0,1)$, con $X=5Z+2$

(1)

$$\mathbb{P}(|X-2|\geq 7)=\mathbb{P}(|5Z|\geq 7)=\mathbb{P}(Z\leq -\frac{7}{5})+\mathbb{P}(Z\geq \frac{7}{5})=$$

$$=^{(simmetria)}2\cdot \mathbb{P}(Z\geq \frac{7}{5})=2\cdot[1-\mathbb{P}(Z\lt \frac{7}{5})]=^{(tavola)}2\cdot [1-0.9192]=0.1616$$

(2)

$$\mathbb{P}(0\leq X\leq 7)=\mathbb{P}(-\frac{2}{5}\leq Z\leq 1)=\mathbb{P}(Z\leq 1)-\mathbb{P}(Z<-\frac{2}{5})=$$

$$=0.8413-[1-\mathbb{P}(Z\geq \frac{2}{5})]=0.8413-[1-0.6554]=0.4967$$

(3)

$$\mathbb{P}(X\geq \alpha)=1-\mathbb{P}(Z\lt \frac{\alpha-2}{5})\leq 0.1\longrightarrow \mathbb{P}(Z\lt \frac{\alpha-2}{5})\geq 0.9$$

$$\mathbb{P}(Z\leq 1.29)\geq 0.9\ \ \ \ \ \ \ \ \ \ \frac{\alpha-2}{5}=1.29\to \alpha=8.45$$

Quindi $\mathbb{P}(X\geq \alpha)\leq 0.1$ se prendiamo $\alpha \geq 8.45$