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Math Help - Probability of a random variable

  1. #1
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    Probability of a random variable

    The reading given by a thermometer calivrated in ice water is a random variable with probablility density function

    f(x) { k(1-x) -1

    { 0 otherwise where k is a constant

    find the value of k.

    What is the probablity that the temperatiure reading is greater than 0C?

    What is the mean reading and the standard devation?
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  2. #2
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    Quote Originally Posted by jabba1 View Post
    The reading given by a thermometer calivrated in ice water is a random variable with probablility density function

    f(x) { k(1-x) -1

    { 0 otherwise where k is a constant

    find the value of k.

    What is the probablity that the temperatiure reading is greater than 0C?

    What is the mean reading and the standard devation?
    Since I can read your density function to find k you need to solve this integral

    \int_{a}^{b}f(x)dx=1 the limits will be the where f(x) is not zero.

    the mean

    \mu=\int_{a}^{b}xf(x)dx

    \sigma^2=\int_{a}^{b}(x-\mu)^2f(x)dx
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    i just noticed a mistake...it should say f(x) { k(1-x) -1<x<1
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    Quote Originally Posted by jabba1 View Post
    i just noticed a mistake...it should say f(x) { k(1-x) -1<x<1
    Well then use that: a=-1~\&~b=1
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  5. #5
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    Quote Originally Posted by jabba1 View Post
    i just noticed a mistake...it should say f(x) { k(1-x) -1<x<1
    okay my original post still stands you need to solve this

    \displaystyle \int_{-1}^{1}k(1-x)dx =1 \iff k \int_{-1}^{1}(1-x)dx=1
    \displaystyle  k\left[ -\frac{(1-x)^2}{2}\bigg|_{-1}^{1}\right]=1 \iff 2k=1 \iff k=\frac{1}{2}

    Now f(x)=\begin{cases} \frac{1-x}{2} , \text{ if }x \in[-1,1] \\ 0, \text{ otherwise }\end{cases}

    Now use this f(x) and the same limits of integration to compute the other 2 integrals.
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    I'm still a little confused on how exactly to use the equation. I think that's my biggest problem
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    Quote Originally Posted by jabba1 View Post
    I'm still a little confused on how exactly to use the equation. I think that's my biggest problem
    \displaystyle \mu=\int_{-1}^{1}x\left(\frac{1-x}{2} \right)dx=\frac{1}{2}\int_{-1}^{1}x-x^2dx

    Then use this for the 2nd one!
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  8. #8
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    thank you! i plugged in the number and for the first one i got .5 and for the second i got .45
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