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    For a continuous random variable why is d(f(x))/dx = f(x) true?

    Could someone explain to me please?
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    It should be d(F(x))/dx = f(x).

    As F(x) = \int_{-\infty}^{x} f(x) dx, thus \frac{d(F(x))}{dx} = f(x).

    Differentiation is the opposite of integration.
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    (Original post by morgan8002)
    It should be d(F(x))/dx = f(x).

    As F(x) = \int_{-\infty}^{x} f(x) dx, thus \frac{d(F(x))}{dx} = f(x).

    Differentiation is the opposite of integration.
    Yes but is that only true for the cumulative distribution function?
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    (Original post by Zenarthra)
    Yes but is that only true for the cumulative distribution function?
    I'm not sure what you mean. F(x) is defined as the cumulative distribution function as it is the integral from -infinity to x of the pdf.
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    (Original post by morgan8002)
    I'm not sure what you mean. F(x) is defined as the cumulative distribution function as it is the integral from -infinity to x of the pdf.
    So when it says the continuous random variable X has a cumulative distribution function F(x) defined by such and such.
    It means this?
    F(x) = \int_{-\infty}^{x} f(x) dx, thus \frac{d(F(x))}{dx} = f(x).

    But why when f(x) is defined on the range of values a<x<b
    then F(x) = INT(f(x))dx from a to x?
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    (Original post by Zenarthra)
    So when it says the continuous random variable X has a cumulative distribution function F(x) defined by such and such.
    It means this?
    F(x) = \int_{-\infty}^{x} f(x) dx, thus \frac{d(F(x))}{dx} = f(x).

    But why when f(x) is defined on the range of values a<x<b
    then F(x) = INT(f(x))dx from a to x?
    Yes, this is what the cumulative distribution function means, it is the integral of f(x) from a to x. Differentiating gives you f(x) back.


    f(x) gives the probability density at any point between a and b.
    Integrating from a to x gives the probability that X is less than x. Thus this is the cumulative distribution function or F(x).
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    (Original post by Zenarthra)
    So when it says the continuous random variable X has a cumulative distribution function F(x) defined by such and such.
    It means this?
    F(x) = \int_{-\infty}^{x} f(x) dx, thus \frac{d(F(x))}{dx} = f(x).

    But why when f(x) is defined on the range of values a<x<b
    then F(x) = INT(f(x))dx from a to x?
    The cdf is calculating the area under the curve from a to x. So when we do the integration, we want the top value to be x. If the top value was b, we would just get the function to be 1 which is incorrect. (1 corresponds to the total area under the curve)

    The cdf is still defined on the range of values a<x<b.
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    Actually F(b) = 1 is correct. If f(x) = 0 for x > b, then P(x<b) = 1.
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    (Original post by morgan8002)
    Actually F(b) = 1 is correct. If f(x) = 0 for x > b, then P(x<b) = 1.
    (Original post by rayquaza17)
    The cdf is calculating the area under the curve from a to x. So when we do the integration, we want the top value to be x. If the top value was b, we would just get the function to be 1 which is incorrect. (1 corresponds to the total area under the curve)

    The cdf is still defined on the range of values a<x<b.
    THanks to the both of you.
    Sorry but why would the area be 1 if we integrated from a to b?
    What is the significance of integrating between a and x?
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    (Original post by Zenarthra)
    THanks to the both of you.
    Sorry but why would the area be 1 if we integrated from a to b?
    What is the significance of integrating between a and x?
    f(x) = 0 outside of a<x<b, so the probability outside a and b is 0.
    As the total probability is 1, the probability between a and b is 1.


    Integrating between a and x gives the probability that X < x.
 
 
 
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