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    Probability distribution of a random variable

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    1.

    Let the random variable X have the p.d.f.

    f(x)=2(1-x) for 0<x<1 and zero elsewhere.

    a. Sketch the graph
    b. Determine and sketch the graph of the distribution function of X
    c. find P(X) for the following intervals: i. [0,1/2] ii. [1/4,3/4] iii. X=3/4 iv. X>3/4

    2.
    For each of the following functions (i) find the constant c so that f(x) is a p.d.f of a random variable X, (ii) find the distribution function F(x)=P(X<x) and (iii) sketch graphs of the p.d.f f(x) and the distribution function F(x):

    a. f(x)=(x^3)/4 0<x<c
    b f(x)=3(x^2)/16 -c<x<c
    c. f(x)= c/sqrt(x) 0<x<1

    3.
    For each of the following functions (i) find the constant c so that f(x) is a p.d.f of a random variable X, (ii) find the distribution function F(x)=P(X<x) and (iii) sketch graphs of the p.d.f f(x) and the distribution function F(x):

    a. f(x)=4x^c 0<x<1
    b. f(x)=c/sqrt(x) 0<x<4
    c. c/(x^3/4) 0<x<1

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    https://brainmass.com/statistics/probability/probability-distribution-of-a-random-variable-259672

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    The 17 page solution file include full derivations and the required graphs.

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