wldnrp13579 said:Homework Statement
fx(X) ={ 1/4 0 < x < 1,
{ 3/8 3 < x < 5,
{ 0 otherwise
Let Y = 1/X. Find the probability density function fy (y) for Y .
Homework Equations
The Attempt at a Solution
Fy(x)=P(Y<x)=P(1/X<x)=P(X>1/x)..
i can't go over more than this
The p.d.f. (probability density function) of x represents the probability of a continuous random variable x taking on a specific value. The p.d.f. of y, on the other hand, represents the probability of a transformed variable y taking on a specific value. In other words, the p.d.f. of y is derived from the p.d.f. of x through a mathematical transformation.
To find the p.d.f. of y from the p.d.f. of x, you must first determine the transformation function that relates x and y. This can be done by solving for y in terms of x. Once the transformation function is known, you can apply the change of variables formula to obtain the p.d.f. of y in terms of the p.d.f. of x.
Finding the p.d.f. of y from the p.d.f. of x is useful in situations where you need to analyze a transformed variable y instead of the original variable x. This is often seen in statistical and scientific research, where data may need to be transformed in order to meet certain assumptions or to make the analysis more meaningful.
Yes, the p.d.f. of y can be found for any transformation of x, as long as the transformation function is one-to-one and differentiable. This means that the transformation must be invertible and have a continuous derivative. If these conditions are met, the p.d.f. of y can be derived using the change of variables formula.
The main limitation is that the transformation function must be known or able to be determined. In some cases, the transformation may be too complex or unknown, making it impossible to find the p.d.f. of y from the p.d.f. of x. Additionally, the transformation may not meet the necessary conditions for the change of variables formula to be applied.