Probability distribution of function of continuous random variables

[kippers]

I hope someone can help me understand functions of random variables:

If X~Uniform(A,B), A < X < B
Y~Normal(0,1), -inf < Y < inf
and Z = X + Y

- what is the pdf of Z?
- how can I calculate a probability like P(Z < 3)?
- what is the conditional probability P(Z<z | X = x)?
- what is the conditional pdf of Z given X = x?

Any hint will be appreciated. Thank you.

 

[TenaliRaman]

What have u tried for 1?

-- AI

 

[tacman]

The probability distribution of a function of continuous random variables can be found using the transformation method. In this case, we have two random variables X and Y, and we are interested in finding the probability distribution of their sum, Z = X + Y.

To find the pdf of Z, we can use the following formula:

fZ(z) = ∫fX(x)fY(z-x)dx

Where fX(x) and fY(y) are the pdfs of X and Y respectively. In this case, fX(x) = 1/(B-A) and fY(y) = 1/(√(2π))e^(-y^2/2).

Substituting these values into the formula, we get:

fZ(z) = ∫1/(B-A) * 1/(√(2π))e^(-(z-x)^2/2)dx

To calculate a probability like P(Z < 3), we can use the pdf of Z and integrate from -∞ to 3. This will give us the area under the curve from -∞ to 3, which represents the probability of Z being less than 3.

To calculate the conditional probability P(Z < z | X = x), we can use the conditional probability formula:

P(Z < z | X = x) = P(Z < z, X = x) / P(X = x)

We can find the joint probability P(Z < z, X = x) by using the joint pdf of X and Y, which is given by fX(x)fY(y). We can then integrate over the range of y from -∞ to z-x and divide by P(X = x), which can be found by integrating fX(x) over the range of X.

The conditional pdf of Z given X = x can be found by using the conditional probability formula and differentiating with respect to z. This will give us the conditional pdf of Z given X = x, which represents the probability distribution of Z when X takes on a specific value x.

I hope this helps in understanding the concept of functions of random variables. Remember to always check the ranges of the random variables and use the appropriate pdfs when finding probabilities or conditional probabilities.