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To find the marginal probability density functions of X and Y, we need to integrate the joint probability density function f(x,y) over the other variable.
First, let's find the marginal probability density function of X:

f_{X} (x) = \int_{- \infty}^{\infty} f (x, y) d y

Since f(x,y) is zero for y outside the range [0,1], we can rewrite the above integral as:

f_{X} (x) = \int_{0}^{1} f (x, y) d y

Using the definition of f(x,y), we have:

f_{X} (x) = \int_{0}^{1} \frac{2}{3} (x + 2 y) d y

Evaluating the integral, we get:

f_{X} (x) = \frac{2}{3} x + \frac{1}{3}

for

0 \leq x \leq 1

, and

f_{X} (x) = 0

elsewhere.
Next, let's find the marginal probability density function of Y:

f_{Y} (y) = \int_{- \infty}^{\infty} f (x, y) d x

Again, since f(x,y) is zero for x outside the range [0,1], we can rewrite the above integral as:

f_{Y} (y) = \int_{0}^{1} f (x, y) d x

Using the definition of f(x,y), we have:

f_{Y} (y) = \int_{0}^{1} \frac{2}{3} (x + 2 y) d x

Evaluating the integral, we get:

f_{Y} (y) = \frac{4}{3} y + \frac{1}{3}

for

0 \leq y \leq 1

, and

f_{Y} (y) = 0

elsewhere.
Therefore, the marginal probability density function of X is:

f_{X} (x) = {\begin{matrix} \frac{2}{3} x + \frac{1}{3} & 0 \leq x \leq 1 \\ 0 & elsewhere \end{matrix}

and the marginal probability density function of Y is:

f_{Y} (y) = {\begin{matrix} \frac{4}{3} y + \frac{1}{3} & 0 \leq y \leq 1 \\ 0 & elsewhere \end{matrix}

Store	Original Price of Aquarium ($)
Pets Plus	118
Pet Planet	110

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