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Can a random variable be constant?
In probability theory, a constant random variable is a discrete random variable that takes a constant value, regardless of any event that occurs. This is technically different from an almost surely constant random variable, which may take other values, but only on events with probability zero.
Are random variables and constants independent?
constants are independent of everything. no non-constant random variable is independent from itself. variance of the sum of independent random variables is the sum of the variances.
Is the mean of a random variable a random variable?
The expected value, or mean, of a random variable—denoted by E(x) or μ—is a weighted average of the values the random variable may assume. In the discrete case the weights are given by the probability mass function, and in the continuous case the weights are given by the probability density function.
Why is the mean of a random variable important?
Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.
Can a non centered random variable be sub Gaussian?
In some sense, you can still get a useful notion for non-centered random variables to be sub-Gaussian, but only if you change the definition.
How does the law of large numbers affect random variables?
The law of large numbersstates that the observed random mean from an increasingly large number of observations of a random variable will always approach the distribution mean . That is, as the number of observations increases, the mean of these observations will become closer and closer to the true mean of the random variable.
Why are variances added to sum and difference of random variables?
Variances are added for both the sum and difference of two independent random variables because the variation in each variable contributes to the variation in each case. If the variables are not independent, then variability in one variable is related to variability in the other.