2.5.8 The random variable X measures the concentration of ethanol in a chemical solution, and the random variable Y measures the acidity of the solution. They have a joint probability density function f(x, y) — А(20 -х — 2у) for 0

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2.5.8 The random variable X measures the concentration of
ethanol in a chemical solution, and the random variable Y
measures the acidity of the solution. They have a joint
probability density function

f(x, y) = A(20 - x - 2y)

2.5.8 The random variable X measures the concentration of
ethanol in a chemical solution, and the random variable Y
measures the acidity of the solution. They have a joint
probability density function
f(x, y) = A(20 – x – 2y)
for 0 <x < 5 and 0 < y < 5, and f (x, y) = 0
elsewhere.
(a) What is the value of A?
(b) What is P(1 < X < 2, 2 < Y < 3)?
(c) Construct the marginal probability density functions
fx(x) and fr(y).
(d) Are the ethanol concentration and the acidity
independent?
(e) What are the expectation and the variance of the
ethanol concentration?
(f) What are the expectation and the variance of the
acidity?
(g) If the ethanol concentration is 3, what is the
conditional probability density function of the
acidity?
Transcribed Image Text:2.5.8 The random variable X measures the concentration of ethanol in a chemical solution, and the random variable Y measures the acidity of the solution. They have a joint probability density function f(x, y) = A(20 – x – 2y) for 0 <x < 5 and 0 < y < 5, and f (x, y) = 0 elsewhere. (a) What is the value of A? (b) What is P(1 < X < 2, 2 < Y < 3)? (c) Construct the marginal probability density functions fx(x) and fr(y). (d) Are the ethanol concentration and the acidity independent? (e) What are the expectation and the variance of the ethanol concentration? (f) What are the expectation and the variance of the acidity? (g) If the ethanol concentration is 3, what is the conditional probability density function of the acidity?
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