(1) Find the density function of Z=max{X, Y};

For Z=max{X, Y}, we can calculate its cumulative distribution function to solve its density function.

First, we can calculate the CDF of z, that is, P(Z≤z).

When z

When 0≤z≤ 1, P(Z≤z)=P(max{X, Y}≤z).

According to the nature of the maximum, we can get the following two situations:

When z≤ 1, P(max{X, Y}≤z)=P(X≤z, Y≤z).

Since x and y are independent, we can decompose their joint probability density function into the product of marginal probability density function:

P(X≤z，Y≤z)=P(X≤z)P(Y≤z).

According to the given density function, we can calculate the marginal probability density function:

P(X≤z)=∫[0，z]∫[z， 1]f(x，y)dydx .

P(Y≤z)=∫[0，z]∫[z， 1]f(x，y)dxdy .

Substituting f(x, y) into the above integral formula, we can calculate P(X≤z) and P(Y≤z).

When z> is 1, P(max{X, Y}≤z)= 1, because the range of z is non-negative.

To sum up, we can get the CDF of z:

F(z) = { 0，z & lt0 P(X≤z)P(Y≤z)，0 ≤ z ≤ 1 1，z & gt 1 }

Then, we can take the derivative of CDF and get the density function of Z.

f(z) = dF(z)/dz

For 0 ≤ z ≤ 1, we can calculate f(z) as follows:

f(z) = d/dz [P(X≤z)P(Y≤z)]

For z > 1, f(z) = 0.

To sum up, the density function of z is:

f(z) = { ( 1-z)e^z，0 ≤ z ≤ 1 0，z & gt 1 }

(2) Find the density function of Z=min{X, Y}:

For Z=min{X, Y}, we can calculate its cumulative distribution function to solve its density function.

First, we can calculate the CDF of z, that is, P(Z≤z).

When z

When 0≤z≤ 1, P(Z≤z)=P(min{X, Y}≤z).

According to the nature of the minimum, we can get the following two situations:

When z≤ 1, p (min {x, y} ≤ z) =1-p (x >; z，Y & gtz).

Since x and y are independent, we can decompose their joint probability density function into the product of marginal probability density function:

p(X & gt； z，Y & gtz)= P(X & gt； z)P(Y & gt； z).

According to the given density function, we can calculate the marginal probability density function:

p(X & gt； z)= 1-P(X≤z)= 1-∫[0，z]∫[z， 1]f(x，y)dydx .

p(Y & gt； z)= 1-P(Y≤z)= 1-∫[0，z]∫[z， 1]f(x，y)dxdy .

Substituting f(x, y) into the above integral formula, we can calculate p (x >; Z) and p (y > z).

When z> is 1, P(min{X, Y}≤z)= 1, because the range of z is non-negative.

To sum up, we can get the CDF of z:

F(z) = { 0，z & lt0 1-P(X & gt； z)P(Y & gt； z)，0 ≤ z ≤ 1 1，z & gt 1 }

Then, we can take the derivative of CDF and get the density function of Z.

f(z) = dF(z)/dz

For 0 ≤ z ≤ 1, we can calculate f(z) as follows:

f(z)= d/dz[ 1-P(X & gt； z)P(Y & gt； z)]

For z > 1, f(z) = 0.

To sum up, the density function of z is:

f(z) = { ( 1-z)e^z，0 ≤ z ≤ 1 0，z & gt 1 }