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The vertical distribution of some marine invertebrates follows a probability distribution with depth p(x) = 2Bxe-B, where B > 0. This means that the probability that any particular individual invertebrate is found between depths x and x + Ax meters is approximately p (x) Ax. (a) Show that the depth at which the probability density function is maxi- mized (e.g., the depth at which there is the highest probability of finding an invertebrate) is x = (26)¯2. -1/2 (b) Multiplying the density function by a constant a gives a function that provides the organism density S(x) = ap (x) at each depth (the units for a will be individuals per cubic meter). Find a and ß so that the maximum density of 115 organisms per cubic meter occurs at a depth of 35 meters. (c) Using the values of a and ß found above, find the total number of inver- tebrates between x = 0 and x = 50 meters. Hint: Take the integral of S (x) to find this.

The vertical distribution of some marine invertebrates follows a probability distribution with depth p(x) = 2Bxe-B, where B > 0. This means that the probability that any particular individual invertebrate is found between depths x and x + Ax meters is approximately p (x) Ax. (a) Show that the depth at which the probability density function is maxi- mized (e.g., the depth at which there is the highest probability of finding an invertebrate) is x = (26)¯2. -1/2 (b) Multiplying the density function by a constant a gives a function that provides the organism density S(x) = ap (x) at each depth (the units for a will be individuals per cubic meter). Find a and ß so that the maximum density of 115 organisms per cubic meter occurs at a depth of 35 meters. (c) Using the values of a and ß found above, find the total number of inver- tebrates between x = 0 and x = 50 meters. Hint: Take the integral of S (x) to find this.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.1: Continuous Probability Models
Problem 35E
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The vertical distribution of some marine invertebrates follows a probability distribution with depth p(x) = 2Bxe-Bx, where B > 0. This means that the probability that any particular individual invertebrate is found between depths x and x + Ax meters is approximately p (x) Ax. (a) Show that the depth at which the probability density function is maxi- mized (e.g., the depth at which there is the highest probability of finding an invertebrate) is x = (28)¯12. (b) Multiplying the density function by a constant a gives a function that provides the organism density S(x) = ap (x) at each depth (the units for a will be individuals per cubic meter). Find a and ß so that the maximum density of 115 organisms per cubic meter occurs at a depth of 35 meters. (c) Using the values of a and ß found above, find the total number of inver- tebrates between x = 0 and x = 50 meters. Hint: Take the integral of S (x) to find this.
Transcribed Image Text:The vertical distribution of some marine invertebrates follows a probability distribution with depth p(x) = 2Bxe-Bx, where B > 0. This means that the probability that any particular individual invertebrate is found between depths x and x + Ax meters is approximately p (x) Ax. (a) Show that the depth at which the probability density function is maxi- mized (e.g., the depth at which there is the highest probability of finding an invertebrate) is x = (28)¯12. (b) Multiplying the density function by a constant a gives a function that provides the organism density S(x) = ap (x) at each depth (the units for a will be individuals per cubic meter). Find a and ß so that the maximum density of 115 organisms per cubic meter occurs at a depth of 35 meters. (c) Using the values of a and ß found above, find the total number of inver- tebrates between x = 0 and x = 50 meters. Hint: Take the integral of S (x) to find this.
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