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Math Help - completing the square?

  1. #1
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    completing the square?

    Hi, I'm really bad at basic math can someone please help me complete the square ><?

    <br />
\int_{-\infty}^{\infty} \frac{e^{tx}}{\sigma\sqrt{2\Pi}} e^{\frac{-(x-\mu)^2}{2 \sigma{^2}}} dx<br />

    Then i let,  z = \frac{x - \mu}{\sigma}; dz = \frac{1}{\sigma}; x = z\sigma + \mu

    <br />
\int_{-\infty}^{\infty} \frac{e^{tz\sigma + \mu}}{\sqrt{2\Pi}} e^{\frac{z^2}{2}} dz<br />

    hmm! i really don't know what I'm doing but I'll just keep going..
    <br />
\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\Pi}} e^{\frac{z^2+ 2t(z\sigma + \mu)}{2}}e^{\frac{t^2}{2}} dz<br />

    yeahh i really don't know what to do but the answer is meant to be:
     e^{t\mu + \frac{t^2\sigma^2}{2}}

    I know that eventually I'm meant to have
    <br />
\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\Pi}} e^{\frac{-(x-\mu)^2}{2 \sigma{^2}}} dx = 1<br />

    and then have the answer outside this integral, but i don't know how to do it. Please help me
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  2. #2
    MHF Contributor matheagle's Avatar
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  3. #3
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    Quote Originally Posted by Katina88 View Post
    Hi, I'm really bad at basic math can someone please help me complete the square ><?

    <br />
\int_{-\infty}^{\infty} \frac{e^{tx}}{\sigma\sqrt{2\Pi}} e^{\frac{-(x-\mu)^2}{2 \sigma{^2}}} dx<br />

    Then i let,  z = \frac{x - \mu}{\sigma}; dz = \frac{1}{\sigma}; x = z\sigma + \mu

    <br />
\int_{-\infty}^{\infty} \frac{e^{tz\sigma + \mu}}{\sqrt{2\Pi}} e^{\frac{z^2}{2}} dz<br />

    hmm! i really don't know what I'm doing but I'll just keep going..
    <br />
\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\Pi}} e^{\frac{z^2+ 2t(z\sigma + \mu)}{2}}e^{\frac{t^2}{2}} dz<br />

    yeahh i really don't know what to do but the answer is meant to be:
     e^{t\mu + \frac{t^2\sigma^2}{2}}

    I know that eventually I'm meant to have
    <br />
\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\Pi}} e^{\frac{-(x-\mu)^2}{2 \sigma{^2}}} dx = 1<br />

    and then have the answer outside this integral, but i don't know how to do it. Please help me
    I think you missed minus before z^2 / 2 accidently. If plus is here the integral does not exist.
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