What is the formula for calculating fZ|Y(z|y) in terms of fX for continuous variables X and Y?
To calculate the conditional probability density function (PDF) fZ|Y(z|y) in terms of fX for continuous variables X and Y, we can use the formula for conditional probability density:
fZ|Y(z|y) = fX,Y(z, y) / fY(y)
where fX,Y(z, y) is the joint probability density function (PDF) of X and Y, and fY(y) is the marginal probability density function (PDF) of Y.
To obtain the joint PDF fX,Y(z, y), you will need the joint distribution of X and Y. This can be determined using either experimental data or mathematical models, depending on the context of your problem.
Once you have the joint PDF fX,Y(z, y), you can compute the marginal PDF fY(y) by integrating the joint PDF over the range of X:
fY(y) = ∫ fX,Y(z, y) dz
The integral represents the summation of the joint probabilities over all possible values of X for a fixed value of Y.
Once you have both fX,Y(z, y) and fY(y), you can then substitute these values into the formula:
fZ|Y(z|y) = fX,Y(z, y) / fY(y)
to calculate the conditional PDF fZ|Y(z|y) in terms of fX.
Remember to ensure that your PDFs are properly normalized and valid in their respective ranges.
To calculate fZ|Y(z|y) in terms of fX for continuous variables X and Y, you can use the conditional probability density function formula.
The formula for fZ|Y(z|y) is derived using the joint probability density function (PDF) of X and Y, denoted as fXY(x,y), and the marginal PDFs of X and Y, denoted as fX(x) and fY(y).
The formula is as follows:
fZ|Y(z|y) = fXY(z, y) / fY(y)
Where:
- fZ|Y(z|y) is the conditional probability density function of Z given Y.
- fXY(x, y) is the joint PDF of X and Y.
- fY(y) is the marginal PDF of Y.
However, this formula assumes that the variables X and Y are continuous and that the joint and marginal PDFs are available. If you have specific values for X, Y, and Z, you can substitute them into the formula to calculate the conditional probability density.