We call p(x, y) the probability density function for attributes x and y if the probability that an arbitrary member of a population has attribute x over the interval (a, b) as well as attribute y over the interval (c, d), corresponds to the volume under the graph of p(x, y) over the rectangle a ≤ x ≤ b, c ≤ y ≤ d, where the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y. (a) Explain why the two conditions given at the end: “the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y” are essential in the context of probabilities. Explain in simple terms (b) Express the probability (volume) referred to in the statement above as a double integral. Intervals should correspond to the region of integration and the order of integration are consistent and clearly indicated. (c) Use answer in part (b) with p(x, y) = (see image) to find 1. Verify that p(x, y) satisfies the definition (given above) of a probability density function. 2. the probability that x > 1/3. Show all steps 3. the probability that x < (1/3) + y. Show all steps.
We call p(x, y) the probability density function for attributes x and y if the probability that an arbitrary member of a population has attribute x over the interval (a, b) as well as attribute y over the interval (c, d), corresponds to the volume under the graph of p(x, y) over the rectangle a ≤ x ≤ b, c ≤ y ≤ d, where the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y. (a) Explain why the two conditions given at the end: “the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y” are essential in the context of probabilities. Explain in simple terms (b) Express the probability (volume) referred to in the statement above as a double integral. Intervals should correspond to the region of integration and the order of integration are consistent and clearly indicated. (c) Use answer in part (b) with p(x, y) = (see image) to find 1. Verify that p(x, y) satisfies the definition (given above) of a probability density function. 2. the probability that x > 1/3. Show all steps 3. the probability that x < (1/3) + y. Show all steps.
We call p(x, y) the probability density function for attributes x and y if the probability that an arbitrary member of a population has attribute x over the interval (a, b) as well as attribute y over the interval (c, d), corresponds to the volume under the graph of p(x, y) over the rectangle a ≤ x ≤ b, c ≤ y ≤ d, where the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y. (a) Explain why the two conditions given at the end: “the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y” are essential in the context of probabilities. Explain in simple terms (b) Express the probability (volume) referred to in the statement above as a double integral. Intervals should correspond to the region of integration and the order of integration are consistent and clearly indicated. (c) Use answer in part (b) with p(x, y) = (see image) to find 1. Verify that p(x, y) satisfies the definition (given above) of a probability density function. 2. the probability that x > 1/3. Show all steps 3. the probability that x < (1/3) + y. Show all steps.
We call p(x, y) the probability density function for attributes x and y if the probability that an arbitrary member of a population has attribute x over the interval (a, b) as well as attribute y over the interval (c, d), corresponds to the volume under the graph of p(x, y) over the rectangle a ≤ x ≤ b, c ≤ y ≤ d, where the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y.
(a) Explain why the two conditions given at the end: “the total volume under p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y” are essential in the context of probabilities. Explain in simple terms
(b) Express the probability (volume) referred to in the statement above as a double integral. Intervals should correspond to the region of integration and the order of integration are consistent and clearly indicated. (c) Use answer in part (b) with p(x, y) = (see image)
to find
1. Verify that p(x, y) satisfies the definition (given above) of a probability density function. 2. the probability that x > 1/3. Show all steps 3. the probability that x < (1/3) + y. Show all steps.
Transcribed Image Text:p(x, y)
=
2
(x+2y) for 0≤x≤1, 0 ≤ y ≤1
otherwise
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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