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Thread: Probability density function

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    Newbie feli3105's Avatar
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    Probability density function

    Given the probability density function $f(x)=e^x$ and a mean of 16, find $P(X<20)$.

    I did: $$P(X<20)=\int_{-\infty}^{20}e^{x}dx=e^{x}\bigg\rvert_{-\infty}^{20}\big[e^{20}-e^{-\infty}\big]=e^{20}$$

    But a probability has to be between 0 and 1. I would appreciate some help, thanks.
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    Re: Probability density function

    as written this problem makes no sense. You are missing a scaling constant somewhere.
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    Re: Probability density function

    This is the prompt, I did my best to translate it as accurate as possible.

    It has been proven that the average lifespan of a pacemaker follows an exponential distribution with a mean of 16 years.

    a) Find the probability that a person who has been implanted with this pacemaker would need a replacement before 20 years.

    b) If the peacemaker has worked properly for 5 years, find the probability that it would need to be replaced before 25 years.
    Last edited by feli3105; Nov 10th 2017 at 08:47 PM.
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    Re: Probability density function

    Quote Originally Posted by feli3105 View Post
    This is the prompt, I did my best to translate it as accurate as possible.

    It has been proven that the average lifespan of a pacemaker follows an exponential distribution with a mean of 16 years.

    a) Find the probability that a person who has been implanted with this pacemaker would need a replacement before 20 years.

    b) If the peacemaker has worked properly for 5 years, find the probability that it would need to be replaced before 25 years.
    ok this makes sense. Let $L$ be the pacemaker time until failure.

    $L \sim \lambda e^{-\lambda t},~\lambda = \dfrac {1}{16}$

    a) $P[L < 20] = \displaystyle \int_0^{20} \lambda e^{-\lambda t}~dt= 1-\dfrac{1}{e^{5/4}} \approx 0.713495$

    b) this makes use of the memoryless property of the exponential distribution

    $P[L > s + t | L > s] = P[L > t]$

    $\begin{align*}
    &P[L < 25~ | ~L >5] = \\
    &1 - P[L > 25 ~|~ L > 5]=\\
    &1-P[L > 20 + 5~ |~ L > 5] = \\
    &1 - P[L > 20] = \\
    &P[L < 20]
    \end{align*}$

    $P[L < 20]$ we found in (a)
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    Re: Probability density function

    Thank you so much! Could you please explain to me how did you integrate $\lambda e^{-\lambda t}$?
    Last edited by feli3105; Nov 11th 2017 at 09:36 AM.
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    Re: Probability density function

    Quote Originally Posted by feli3105 View Post
    Thank you so much! Could you please explain to me how did you integrate $\lambda e^{-\lambda t}$?
    What is the derivative of $\large{-e^{-\lambda t}}~?$
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    Re: Probability density function

    Quote Originally Posted by feli3105 View Post
    Thank you so much! Could you please explain to me how did you integrate $\lambda e^{-\lambda t}$?
    come on, it's about as basic an integration as there is.
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    Re: Probability density function

    The thing is I had calculus like 4 years ago and I forgot pretty much about integration and derivatives :/

    I think I'm doing something wrong because I'm not arriving to the same result as you did, could you please check out my integration?

    $$P(X<20)=\int_{0}^{20}\lambda e^{-\lambda t}dt=\lambda \int_{0}^{20}e^{-\lambda t}dt=\frac{\lambda e^{-\lambda t}}{-\lambda}=\bigg[-e^{-\lambda t}\bigg]_{0}^{20}=-\bigg[1-\frac{1}{e^{5/4}}\bigg]=\frac{1}{e^{5/4}}-1$$
    Last edited by feli3105; Nov 11th 2017 at 01:39 PM.
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    Re: Probability density function

    $$\lambda e^{-\lambda t}$$
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    Re: Probability density function

    Quote Originally Posted by feli3105 View Post
    The thing is I had calculus like 4 years ago and I forgot pretty much about integration and derivatives :/

    I think I'm doing something wrong because I'm not arriving to the same result as you did, could you please check out my integration?

    $$P(X<20)=\int_{0}^{20}\lambda e^{-\lambda t}dt=\lambda \int_{0}^{20}e^{-\lambda t}dt=\frac{\lambda e^{-\lambda t}}{-\lambda}=\bigg[-e^{-\lambda t}\bigg]_{0}^{20}=-\bigg[1-\frac{1}{e^{5/4}}\bigg]=\frac{1}{e^{5/4}}-1$$
    starting from

    $\left . -e^{-\lambda t} \right |_0^{20} = \left . e^{-\lambda t} \right |_{20}^0 = 1 - e^{-20\lambda} = 1 - e^{-5/4} = 1 - \dfrac{1}{e^{5/4}}$
    Thanks from feli3105
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    Re: Probability density function

    Quote Originally Posted by feli3105 View Post
    The thing is I had calculus like 4 years ago and I forgot pretty much about integration and derivatives.
    If that statement is true, then you have no business taking a probability course that requires calculus. In the department I retired from, the outline required students to sign a statement to the effect that they considered themselves proficient in calculus, logic & set-theory.
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    Re: Probability density function

    Quote Originally Posted by romsek View Post
    starting from

    $\left . -e^{-\lambda t} \right |_0^{20} = \left . e^{-\lambda t} \right |_{20}^0 = 1 - e^{-20\lambda} = 1 - e^{-5/4} = 1 - \dfrac{1}{e^{5/4}}$
    Perhaps it's easier to remember that the CDF function of an exponential distribution is $1-e^{-\lambda t}$ if $t \geqslant 0$ and $0$ if $t<0$.
    Last edited by feli3105; Nov 11th 2017 at 02:35 PM.
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    Re: Probability density function

    Quote Originally Posted by feli3105 View Post
    Perhaps it's easier to remember that the CDF function of an exponential distribution is $1-e^{-\lambda t}$ if $t \geqslant 0$ and $0$ if $t<0$.
    Memorizing formulas is no way to go. Your brain will run out of room!
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    Re: Probability density function

    Quote Originally Posted by Plato View Post
    If that statement is true, then you have no business taking a probability course that requires calculus. In the department I retired from, the outline required students to sign a statement to the effect that they considered themselves proficient in calculus, logic & set-theory.
    Thank you for your advice Plato, but I have no other alternative than to take this biostatistics course because it's compulsory to earn my bachelor's degree in science and unfortunately it's the only one available. You see, math courses aren't well planned in Uruguay, math professors hate teaching anyone who doesn't study maths because they have to lower the bar. I had calculus in my first semester and this course should have been taught the following semester when calculus was still fresh in my mind. Instead they opt to teach it 3 years later when hardly anyone remembers how to solve an integral!!
    I think I'll have to get over it somehow, sorry for the rant.
    Last edited by feli3105; Nov 11th 2017 at 05:34 PM.
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    Re: Probability density function

    Quote Originally Posted by romsek View Post
    Memorizing formulas is no way to go. Your brain will run out of room!
    You're right! Wouldn't you mind explaining to me what the memoryless property is about and when do I have to use it? It looks like a conditional probability though (because of the "|") but I'm not sure what does $P[L>s+t|L>s]=P[L>t]$ mean.

    You're saving my life romsek, thank you so much!! If you ever need help with biology stuff I'd like to help you out.
    Last edited by feli3105; Nov 11th 2017 at 05:42 PM.
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