4. Let ( X ) be a continuous random variable with a density function, [ f(x)=left{egin{array}{ll} a x^{2} & 0<x<2, \ 0 & ext { otherwise } end{array} ight. ] (a) (5 points) Find a. (b) (5 points) Find the variance of ( X ). (c) (5 points) Find the cumulative distribution function (cdf) of ( X ).
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So, we have: ∫ from 0 to 2 of a*x^2 dx = 1 This integral evaluates to (a/3)*x^3 from 0 to 2, which simplifies to (8a/3) - 0 = 1. Solving for a, we get a = 3/8. (b) The variance of a random variable X is given by E[X^2] - (E[X])^2. Show more…
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