Let X be uniform on [0,1/2] . Find the PDF fY(y) of Y=X/(1−X) .
For 0≤y≤1 :
To find the PDF (Probability Density Function) of Y, we need to determine the cumulative distribution function (CDF) of Y and then differentiate it to find the PDF.
Step 1: Determine the relationship between X and Y.
From the given equation Y = X / (1 - X), we can rearrange it as (1 - X)Y = X, which gives us Y - XY = X.
Now, we can solve for X in terms of Y by factoring out X: X(1 + Y) = Y.
Dividing both sides by (1 + Y), we get X = Y / (1 + Y).
Step 2: Find the cumulative distribution function (CDF) of Y.
To find the CDF of Y, we need to determine the probability that Y is less than or equal to a given value y, denoted as P(Y ≤ y).
For 0 ≤ y ≤ 1, we have:
P(Y ≤ y) = P(X / (1 - X) ≤ y)
= P(Y / (1 + Y) ≤ y)
= P(X ≤ y / (1 + y)), using the relationship X = Y / (1 + Y).
Since X is uniformly distributed on [0, 1/2], we can find the probability by calculating the area under the probability density function within the given interval [0, y / (1 + y)].
P(Y ≤ y) = ∫[0, y / (1 + y)] fX(x) dx
Step 3: Calculate the probability by integrating the PDF of X.
The PDF of X, denoted as fX(x), is simply a constant function equal to 2 within the interval [0, 1/2] since X is uniformly distributed in that range.
For 0 ≤ y ≤ 1, we can evaluate the integral as follows:
P(Y ≤ y) = ∫[0, y / (1 + y)] fX(x) dx
= ∫[0, y / (1 + y)] 2 dx
Integrating this expression will give us the cumulative distribution function (CDF) of Y.
Step 4: Differentiate the CDF to obtain the PDF of Y.
Once we have the CDF of Y, we can differentiate it with respect to y to obtain the PDF of Y, denoted as fY(y).
fY(y) = d/dy [P(Y ≤ y)]
By differentiating the CDF of Y obtained in Step 3, we can find the PDF of Y over the given range 0 ≤ y ≤ 1.
Note: The final steps involve mathematical calculations, and it would be helpful to consult a textbook or use mathematical software to perform the integration and differentiation.