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    Density Function of Random Variables

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    The destiny function for a random variable X is given in terms of constant c.....

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    Solution Preview

    Since the integral of density function over the domain of X is equal to 1, we can find the value of c by
    ∫_(-∞)^∞▒〖δ(x)dx=1〗, or
    ∫_(-∞)^0▒〖ce^x dx+∫_0^∞▒〖ce^(-x) dx〗=1〗
    ce^x |_(-∞)^0-ce^(-x) |_0^∞=1
    So c=1/2
    The distribution function is
    If x<0,
    ∫_(-∞)^x▒〖1/2 e^t dt=1/2 e^t |_(-∞)^x 〗=1/2 e^x
    If x≥0
    ∫_(-∞)^0▒〖1/2 e^t dt+∫_0^x▒〖ce^(-t) dt〗=1/2-1/2 e^(-t) |_0^x 〗=1-1/2 e^(-x)
    In general format, the distribution function is
    F(x)={█(〖1/2 e〗^x, x<0@1-1/2 e^(-x),x≥0)┤
    P(X>1)=1-P(X≤1)=1-F(1)=1-1/2 e^(-1)=1-1/2e
    P(-3<X≤4)=F(4)-F(-3)=1-1/2 e^(-4)-1/2 e^(-3)
    P(X<-2)=F(-2)=1/2 e^(-2)=1/(2e^2 )
    The graph of density function is

    The graph of distribution function is

    Solution Summary

    The density function of random variables is examined.