Is sum discrete or continuous?
A discrete random variable has a countable number of possible values. The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1. A continuous random variable takes on all the values in some interval of numbers.
Which random variables are continuous?
A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile. A continuous random variable is not defined at specific values.
Is the sum of a discrete and a continuous random variable continuous?
If Y is a continuous random variable, then Z := X + Y is a hybrid random variable. As we know the probability density functions of X and Y, we can compute the probability density function of Z. Assuming that X and Y are independent, the probability density function of Z is given by the convolution of the probability density functions f X and f Y
How to find the sum of independent random variables?
We now develop a methodology for finding the PDF of the sum of two independent random variables, when these random variables are continuous with known PDFs. So in that case, Z will also be continuous and so will have a PDF. The development is quite analogous to the one for the discrete case.
How to think of X as a continuous random variable?
Random variable X can be thought of as a continuous random variable with the following probability density function where δ is the Dirac delta function. If Y is a continuous random variable, then Z := X + Y is a hybrid random variable.
How to show the general result of a random variable?
We will show this in the special case that both random variables are standard normal. The general case can be done in the same way, but the calculation is messier. Another way to show the general result is given in Example 10.17. Suppose X and Y are two independent random variables, each with the standard normal density (see Example 5.8).