## Related questions with answers

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

VerifiedSolution A

Solution B

Step 1

1 of 3Given:

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

$\textbf{DISTRUBITION}$ $X$

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 $\textbf{normal distribution with mean 500 and variance 10,000}$.

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