Jointly Distrubuted RW/ Determinin the Borders PROBLEM

[topcu_431]

Question 29. When a current I (measured in amperes) flows through a resistance R (measured in ohms), the power generated is given by W = I ² R (measured in watts). Suppose that I and R are independent random variables with densities

. . . . .\(\displaystyle \begin{array}{lr}f_I (x)\, =\, 6x\, (1\, -\, x) & \mbox{ for }\, 0\, \leq\, x\, \leq\, 1 \\ f_R (x)\, =\, 2x & \mbox{ for }\, 0\, \leq\, x\, \leq\, 1 \end{array}\)

Determine the density function of W.

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Solution 29:

. . . . .\(\displaystyle \displaystyle \begin{align} P\, \left\{\, I^2\, R\, \leq\, w\, \right\}\, &=\, \int_{\substack{x^2 \\ 0 \\ 0}}\, \int_{\substack{y\, \leq\, w \\ x\, \leq\, 1 \\ y\, \leq\, 1}} \, 6x\, (1\, -\, x)\, 2y\, dy\, dx

\\ \\ &=\, \int_0^{\sqrt{ w\,}}\, \int_0^1\, 12x\, (1\, -\, x)\, y\, dy\, dx\, +\, \int_{\sqrt{w\,}}^1\, \int_0^{^w/_{x^2}}\, 12x\, (1\, -\, x)y\, dy\, dx

\\ \\ &=\, 3w\, -\, 2w^{{}^3/_2}\, =\, 6w\, \left(1\, +\, (\log(w))/2\, -\, \sqrt{\strut w\,}\right)

\\ \\ &= \, 4w^{^3/_2}\, -\, 3w\, (1\, +\, \log(w)),\, 0\, <\, w\, < 1 \end{align}\)

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I could not understand that how we determine the borders of integral here ? (I have figured out other operations ) .I have attached both question and solution. Thanks for your help, in advance.

 

[Deleted member 4993]

Question 29. When a current I (measured in amperes) flows through a resistance R (measured in ohms), the power generated is given by W = I ² R (measured in watts). Suppose that I and R are independent random variables with densities

. . . . .\(\displaystyle \begin{array}{lr}f_I (x)\, =\, 6x\, (1\, -\, x) & \mbox{ for }\, 0\, \leq\, x\, \leq\, 1 \\ f_R (x)\, =\, 2x & \mbox{ for }\, 0\, \leq\, x\, \leq\, 1 \end{array}\)

Determine the density function of W.

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Solution 29:

. . . . .\(\displaystyle \displaystyle \begin{align} P\, \left\{\, I^2\, R\, \leq\, w\, \right\}\, &=\, \int_{\substack{x^2 \\ 0 \\ 0}}\, \int_{\substack{y\, \leq\, w \\ x\, \leq\, 1 \\ y\, \leq\, 1}} \, 6x\, (1\, -\, x)\, 2y\, dy\, dx

\\ \\ &=\, \int_0^{\sqrt{ w\,}}\, \int_0^1\, 12x\, (1\, -\, x)\, y\, dy\, dx\, +\, \int_{\sqrt{w\,}}^1\, \int_0^{^w/_{x^2}}\, 12x\, (1\, -\, x)y\, dy\, dx

\\ \\ &=\, 3w\, -\, 2w^{{}^3/_2}\, =\, 6w\, \left(1\, +\, (\log(w))/2\, -\, \sqrt{\strut w\,}\right)

\\ \\ &= \, 4w^{^3/_2}\, -\, 3w\, (1\, +\, \log(w)),\, 0\, <\, w\, < 1 \end{align}\)

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I could not understand that how we determine the borders of integral here ? (I have figured out other operations ) .I have attached both question and solution. Thanks for your help, in advance.

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[topcu_431]

I could not determine the domain of X. My work for determining the domain of y is on the picture that i have attached.Your helps are really important for me , because my exam is tomorrow.:neutral: