Background: a number, e.g. 2 , can be thought of as a trivial random variable that always takes the value 2 . Let x be a number.

Let X be a random variable associated with some probabilistic experiment.

a) Is it always true that X+x is a random variable? Yes or No
b) Is it always true that X−x=0 ? Yes or no

a)yes

b)no

a) Well, if X+x is a random variable, then my favorite number is the square root of negative pizza! But seriously, yes, X+x is indeed a random variable. Adding a constant value to a random variable simply shifts its values without changing its probabilistic nature.

b) Is X−x=0 always true? No, it's not always true unless X happens to be equal to x . Otherwise, the difference between X and x can be any value other than zero. So, unless you're dealing with some special scenario where they are equal, X−x won't be zero. Unless you're talking about my math skills, then yes, the answer is always zero!

a) Yes, it is always true that X+x is a random variable.

Explanation: An operation performed on a random variable, such as adding a number, results in another random variable. The sum of a random variable and a constant still follows the rules of probability and can be considered a random variable itself.

b) No, it is not always true that X−x = 0.
Explanation: X−x is equal to zero only if the value of X is equal to the value of x. However, considering X as a random variable, it can take on different values depending on the outcome of a probabilistic experiment. Thus, X−x is not always equal to zero.

a) Yes, it is always true that X+x is a random variable. To understand why, let's first clarify what a random variable is. A random variable is a variable that takes on different values based on the outcomes of a probabilistic experiment. In other words, it is a function that assigns a real number to each outcome of the experiment.

Now, when we add a constant value (x) to a random variable (X), we are still essentially applying a function to the outcomes of the experiment. The function simply adds the constant value to the original values of X. Therefore, the result, X+x, is still a real-valued function of the experiment outcomes, making it a random variable.

b) No, it is not always true that X-x=0. This is because X and x can have different values. The expression X-x represents the subtraction of the random variable X and the constant value x. Depending on the values of X and x, the result can be non-zero.

For example, if X is a random variable representing the outcome of rolling a fair six-sided die (taking values 1, 2, 3, 4, 5, or 6) and x is the constant value of 2, then X-x can take values of -1, 0, 1, 2, 3, or 4. Thus, it is not always true that X-x equals zero.