Question

Let $M(t) := \max_{0 \leq s \leq t} W(s)$. Find (a) $E[M(t)]$, (b) $P(|W(t)| \leq \alpha)$, where $\alpha \geq 0$, and $E[|W(t)|]$.

Name: let mt max0 leq s leq t ws find a emt b pwt leq alpha where alpha geq 0 and ewt 44808
Uploaded: 2024-07-29T16:20:47-08:00
Duration: 1 min 54 s
Channel: Lien Le
Description: let mt max0 leq s leq t ws find a emt b pwt leq alpha where alpha geq 0 and ewt 44808

          Let $M(t) := \max_{0 \leq s \leq t} W(s)$. Find (a) $E[M(t)]$, (b) $P(|W(t)| \leq \alpha)$, where $\alpha \geq 0$, and $E[|W(t)|]$.

Added by Yolanda H.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Lien Le

07/29/2024

Step 1

Therefore, $E[M(t)] = E[|W(t)|]$. Since $W(t) \sim N(0, t)$, the probability density function of $W(t)$ is given by $$f_{W(t)}(x) = \frac{1}{\sqrt{2\pi t}} e^{-\frac{x^2}{2t}}$$ Then, $$E[|W(t)|] = \int_{-\infty}^{\infty} |x| f_{W(t)}(x) dx = Show more…

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