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Math Help - Probability density functions

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    Probability density functions

    Hi could I please get some urgent help with the following?

    1) Let X be a continuous random variable with probability density function given by:

    fX(x) = 2x^(−3) for x>=1
    _______0 otherwise


    (a) Show that fX(x) is a valid density function.

    (b) Explain why fX(1) = 2 is an acceptable value for the density function when probabilities must be less than 1.

    (c) Find the distribution function for X.

    (f) Solve the equation FX(x) = 0.5 for x. What importance does this value have?
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    Quote Originally Posted by One View Post
    Hi could I please get some urgent help with the following?

    1) Let X be a continuous random variable with probability density function given by:

    fX(x) = 2x^(−3) for x>=1
    _______0 otherwise


    (a) Show that fX(x) is a valid density function.

    Mr F says: Show that {\color{red}\int_{1}^{\infty} 2x^{-3} \, dx = 1} and {\color{red}2x^{-3} \geq 0} for {\color{red} x \geq 1}.

    (b) Explain why fX(1) = 2 is an acceptable value for the density function when probabilities must be less than 1. Mr F says: {\color{red} f_X(1) } does not represent a probability.

    (c) Find the distribution function for X. Mr F says: Use the definition you will have in your class notes or textbook.

    (f) Solve the equation FX(x) = 0.5 for x. What importance does this value have? Mr F says: Equate your answer to (c) to 0.5 and solve for x. This value of x is the median of X.
    Go back and revise the basic definitions.
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