Question

# Problem 3 (8 points) If X is a continuous random variable with densityfunction ox (r),we have Assume that X is uniformly distributed on the interval [0, 10], with density \phi_{X}(x)=\left\{\begin{array}{ll}

\frac{1}{10}, & 0 \leq x \leq 10 \\ 0, & \text { otherwise } \end{array}\right. \text { Find } P[0 \leq x \leq 5] \text {. } ) Assume that X is as in (a). Show that the cumulative distribution is givenby F_{X}(x)=\left\{\begin{array}{cl} 0, & x<0 \\ \frac{x}{10}, & 0 \leq x \leq 10 \\ 1 . & x \geq 10 \end{array}\right. O Assume in general that X has density ox(r) and that Y =f(r)) is a strictly increasing function with inverse g(r) = f(x). The cumulative distribution of X is given byf(X), where F_{X}(x)=\int_{-\infty}^{x} \phi_{X}(x) d x Find anexpression for the cumulative distribution of Y and use this toshow that Y has the density \phi_{Y}(x)=\phi_{X}(g(x)) g^{\prime}(x) ) Now assume that X has density as in (a) and that Y = ex. Use the resultin (c) to find the density of Y on the interval [1, e10].

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