Let X and Y be independently random variables, with X uniformly distributed on [0,1] and Y uniformly distributed on [0,2] . Find the PDF fZ(z) of Z=max{X,Y}
1. For z<0 or z>2 : find fZ(z)
2. For 0≤z≤1 : find fZ(z)
1. 0
2. z
3. 1/2
1. 0
2. 2*z
3. For 1≤z≤2 : find fZ(z)?
3. 0.5
To find the probability density function (PDF) of Z, we need to consider the range for each value of Z and calculate the probability that Z takes on each value within that range.
1. For z < 0 or z > 2:
Since Z represents the maximum of X and Y, if z is less than 0 or greater than 2, this means that neither X nor Y can take on such values. Therefore, fZ(z) = 0 for z < 0 or z > 2.
2. For 0 ≤ z ≤ 1:
To find the PDF in this range, we need to consider the two possible cases:
- Case 1: Z = X
- Case 2: Z = Y
Case 1: Z = X
When Z = X, X must be less than or equal to z since Z is the maximum. Therefore, the probability of Z being equal to X is equal to the cumulative distribution function (CDF) of X at z, which is P(X ≤ z) = z.
Case 2: Z = Y
When Z = Y, Y must be less than or equal to z. Since Y is uniformly distributed on [0,2], the probability of Y being less than or equal to z is P(Y ≤ z) = z/2.
Since X and Y are independent, to find the PDF of the maximum, we take the sum of the probabilities of the two cases:
fZ(z) = P(Z = X) + P(Z = Y) = z + z/2 = (3z)/2 for 0 ≤ z ≤ 1.
Therefore, the PDF of Z, fZ(z), for 0 ≤ z ≤ 1 is (3z)/2.