How to find the PDF of a random variable Y?
Thus, the CDF of Y is given by F Y ( y) = { 0 for y < 0 √ y for 0 ≤ y ≤ 1 1 for y > 1 Note that the CDF is a continuous function of Y, so Y is a continuous random variable. Thus, we can find the PDF of Y by differentiating F Y ( y) , f Y ( y) = F ′ Y ( y) = { 1 2 √ y for 0 ≤ y ≤ 1 0 otherwise
Which is the function of a continuous random variable?
It is usually more straightforward to start from the CDF and then to find the PDF by taking the derivative of the CDF. Note that before differentiating the CDF, we should check that the CDF is continuous. As we will see later, the function of a continuous random variable might be a non-continuous random variable.
How to find the probability density of a random variable?
You might not have been aware of it at the time, but we have already used the distribution function technique at least twice in this course to find the probability density function of a function of a random variable. For example, we used the distribution function technique to show that: \\(Z=\\dfrac{X-\\mu}{\\sigma}\\)
Which is an example of a CDF function?
Let’s look at an example. Let X be a Uniform(0, 1) random variable, and let Y = eX . Find EY. First, note that we already know the CDF and PDF of X. In particular, F X ( x) = { 0 for x < 0 x for 0 ≤ x ≤ 1 1 for x > 1 It is a good idea to think about the range of Y before finding the distribution.
How to find a function of a continuous random variable?
Let X be a continuous random variable with PDF fX(x) = {4×3 0 < x ≤ 1 0 otherwise and let Y = 1 X. Find fY(y). First note that R Y = [ 1, ∞). Also, note that g ( x) is a strictly decreasing and differentiable function on ( 0, 1], so we may use Equation 4.5.
Is the PDF of Y a continuous function?
So we immediately know that F Y ( y) = P ( Y ≤ y) = 0, for y < 1, F Y ( y) = P ( Y ≤ y) = 1, for y ≥ e. since 0 ≤ ln y ≤ 1. The above CDF is a continuous function, so we can obtain the PDF of Y by taking its derivative.