2. The joint density function for random variables X, Y, and Z is
0, otherwise
a) Find the value of C.
b) Find P(X ≤ 1,Y ≤ 1, Z ≤ 1).
Here, "f(x,y,z)" is a probability density function.
So,
a) "\\int_0^2\\int_0^2\\int_0^2 Cxyzdzdydx=1"
"\\implies" "\\int_0^2\\int_0^2\\left\\{\\frac{Cxyz\u00b2}{2}\\right\\}_0^2dydx=1"
"\\implies" "\\int_0^2\\int_0^22Cxydydx=1"
"\\implies" "\\int_0^2\\left\\{Cxy\u00b2\\right\\}_0^2dx=1"
"\\implies" "\\int_0^24Cxdx=1"
"\\implies" "\\left\\{\\frac{4Cx\u00b2}{2}\\right\\}_0^2=1"
"\\implies" "\\left\\{2Cx\u00b2\\right\\}_0^2=1"
"\\implies" "2C(2)\u00b2-2C(0)\u00b2=1"
"\\implies" "8C=1"
"\\implies" "C=\\frac{1}{8}"
b) "P(X\u22641,Y\u22641,Z\u22641)"
"=\\int_0^1\\int_0^1\\int_0^1\\frac{xyz}{8}dzdydx"
"=\\int_0^1\\int_0^1\\left\\{\\frac{xyz\u00b2}{16}\\right\\}_0^1dydx"
"=\\int_0^1\\int_0^1\\frac{xy}{16}dydx"
"=\\int_0^1\\left\\{\\frac{xy\u00b2}{32}\\right\\}_0^1dx"
"=\\int_0^1\\frac{x}{32}dx"
"=\\left\\{\\frac{x\u00b2}{64}\\right\\}_0^1=\\frac{1}{64}"
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