Try the fastest way to create flashcards
Question

# If the moment-generating function of X is given by$M(t) = e^{500t + 5000t^2}$, find P[27, 060 ≤ (X - 500)² ≤ 50,240].

Solutions

Verified
Step 1
1 of 3

Given:

$M(t)=\exp(500t+5000t^2)$

$\textbf{DISTRUBITION}$ $X$

Moment generating function of the normal distribution:

$M(t)=\exp\left(\mu y+\dfrac{\sigma^2 t^2}{2}\right)$

We then note that the moment generating function $\exp(500t+5000t^2)$ has the form of the moment generating function of the normal distribution with:

$\mu=500$

$\dfrac{\sigma^2}{2}=5000\Rightarrow \sigma^2=2\cdot 5000=10,000$

We then note that the $X$ has a $\textbf{normal distribution with mean 500 and variance 10,000}$.

## Recommended textbook solutions

#### Probability and Statistical Inference

9th EditionISBN: 9780321923271Dale Zimmerman, Elliot Tanis, Robert V. Hogg
Verified solutions

#### Probability and Statistics for Engineering and the Sciences

8th EditionISBN: 9780538733526 (4 more)Jay L. Devore
1,232 solutions

#### Probability and Statistics for Engineering and the Sciences

9th EditionISBN: 9781305251809 (9 more)Jay L. Devore
1,589 solutions

#### Statistics and Probability with Applications

3rd EditionISBN: 9781464122163Daren S. Starnes, Josh Tabor
2,555 solutions