Question

QUESTION1 acidity, is important in studies of acid rain. For a certain Florida lake, baseline measurements of acidity are made so that any changes caused by acid rain can be noted. The pH for water The pH level, samples from the lake is a random variable X, with probability density function fx(x)- elsewhere What is the value of a? (Report answer to three decimal places Answer musf be within0.001 of correct value.) QUESTION 2 The pH level, a measure of acidity, is important in studies of acid rain. For a certain Florida lake, baseline measurements of acidity are made so that any changes caused by acid rain can be noted. The pH for water samples from the lake is a random variable X, with probability density function fx(x)-64 elsewhere Which of the following is the cumulative distribution function for X? Hint: The possible answers differ only in the intervals used to define the piecewise function. For instance, one option has 3 as one of the intervals while another has x 3.) 0 64 xS3 64 XS 3 64 7s X O All of the above

Answer 1

Question 1:

For a valid pdf following must be true:

$\int_{3}^{7}f(x)dx=1$ .............(1)

Now,

$\int_{4}^{7}f(x)dx=\int_{4}^{7}a(7-x)^{2}dx=\int_{4}^{7}a\left [ 49+x^{2}-14x \right ]dx=a\left [ 49x+\frac{x^{3}}{3}-7x^{2} \right ]_{4}^{7}=9a$

............(2)

Equating equation (1) and (2) gives

a = 1/9 = 0.111

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Question 2:

$F(x)=\int_{3}^{x}f(x)dx=\frac{3}{64}\int_{3}^{x}(7-x)^{2}dx=\frac{3}{64}\int_{3}^{x}\left [ 49+x^{2}-14x \right ]dx=\frac{3}{64}\left [ 49x+\frac{x^{3}}{3}-7x^{2} \right ]_{3}^{x}=\frac{3}{64}\left [ 49x+\frac{x^{3}}{3}-7x^{2}-93 \right ]=1-\frac{1}{64}(7-x)^{3}$

First option is correct.