Let 𝑋 be uniform on [0,1/2] . Find the PDF 𝑓𝑌(𝑦) of 𝑌=𝑋/(1−𝑋) .
For 0≤𝑦≤1, what is 𝑓𝑌(𝑦) ?
To find the PDF of Y, we first need to find the cumulative distribution function (CDF) of Y and then differentiate it to obtain the PDF.
Let's start by finding the CDF of Y.
The definition of the CDF of a random variable Y is given by:
F_Y(y) = P(Y ≤ y)
Since Y = X / (1 - X), we can rewrite the above equation as:
F_Y(y) = P(X / (1 - X) ≤ y)
Multiplying both sides of the inequality by (1 - X), we get:
F_Y(y) = P(X ≤ y * (1 - X))
Now, since X is uniformly distributed on [0, 1/2], its PDF is constant over this interval and equal to 2. Therefore, we can rewrite the above equation as an integral:
F_Y(y) = ∫[0, y*(1 - x)] 2 dx
Integrating with respect to x, we get:
F_Y(y) = 2∫[0, y*(1 - x)] dx
F_Y(y) = 2[x]_[0, y*(1 - x)]
F_Y(y) = 2[y*(1 - y) - 0]
F_Y(y) = 2y(1 - y)
Now, to find the PDF f_Y(y), we differentiate the CDF with respect to y:
f_Y(y) = d/dy [F_Y(y)]
f_Y(y) = d/dy [2y(1 - y)]
f_Y(y) = 2(1 - 2y)
Therefore, the PDF of Y, f_Y(y), for 0 ≤ y ≤ 1 is given by:
f_Y(y) = 2(1 - 2y)
To find the PDF of 𝑌, you need to follow these steps:
Step 1: Determine the cumulative distribution function (CDF) of 𝑌, 𝐹𝑌(𝑦).
Step 2: Differentiate the CDF to obtain the PDF, 𝑓𝑌(𝑦).
Let's find the CDF and then differentiate it to find the PDF.
Step 1: Calculate the CDF, 𝐹𝑌(𝑦):
To find the CDF of 𝑌, we need to find the probability that 𝑌 takes on a value less than or equal to 𝑦 (i.e., 𝑃(𝑌 ≤ 𝑦)).
Since 𝑌 = 𝑋 / (1 - 𝑋), we can write 𝑌(1 - 𝑋) = 𝑋.
Expanding the equation gives 𝑌 - 𝑌𝑋 = 𝑋, and rearranging gives 𝑋 = 𝑌 / (1 + 𝑌).
Now, we can express the inequality 𝑌 ≤ 𝑦 in terms of 𝑋:
𝑌 ≤ 𝑦
⇒ 𝑌 / (1 + 𝑌) ≤ 𝑦.
Solving this inequality for 𝑋, we get:
𝑋 ≤ 𝑌 / (1 + 𝑌).
Since 𝑋 is uniform on [0, 1/2], the probability that 𝑋 ≤ 𝑦 / (1 + 𝑦) is simply 𝑦 / (1 + 𝑦).
Therefore, the CDF of 𝑌, 𝐹𝑌(𝑦), is given by:
𝐹𝑌(𝑦) = 𝑦 / (1 + 𝑦), for 0 ≤ 𝑦 ≤ 1.
Step 2: Differentiate the CDF to obtain the PDF, 𝑓𝑌(𝑦):
To find the PDF, we differentiate the CDF with respect to 𝑦.
Differentiating 𝐹𝑌(𝑦) = 𝑦 / (1 + 𝑦) with respect to 𝑦 gives:
𝑓𝑌(𝑦) = d/d𝑦 (𝑦 / (1 + 𝑦)).
To differentiate this expression, we can use the quotient rule:
𝑓𝑌(𝑦) = [(1 + 𝑦) * d(𝑦) / d𝑦 - 𝑦 * d(1 + 𝑦) / d𝑦] / (1 + 𝑦)².
Simplifying, we have:
𝑓𝑌(𝑦) = (1 / (1 + 𝑦)²).
Therefore, the PDF of 𝑌, 𝑓𝑌(𝑦), for 0 ≤ 𝑦 ≤ 1 is:
𝑓𝑌(𝑦) = 1 / (1 + 𝑦)².