Contents
How do you find the expectation of a continuous function?
μ=μX=E[X]=∞∫−∞x⋅f(x)dx. The formula for the expected value of a continuous random variable is the continuous analog of the expected value of a discrete random variable, where instead of summing over all possible values we integrate (recall Sections 3.6 & 3.7).
What is conditional expectation function?
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of “conditions” is known to occur.
What is the expectation of a constant?
The expected value of a constant is just the constant, so for example E(1) = 1. Multiplying a random variable by a constant multiplies the expected value by that constant, so E[2X] = 2E[X].
How do you do conditional expectations?
The conditional expectation, E(X |Y = y), is a number depending on y. If Y has an influence on the value of X, then Y will have an influence on the average value of X. So, for example, we would expect E(X |Y = 2) to be different from E(X |Y = 3).
Which is the variance of the RV X?
The variance measures the amount of variability of the RV X around E ( X). Definition 2.3.2. The variance of an RV X is the expectation of the RV Y = ( X − E ( X)) 2: V a r ( X) = E ( ( X − E ( X)) 2). The standard deviation of an RV X is s t d ( X) = V a r ( X) .
What is the expectation of a Cauchy random variable?
A Cauchy random variable takes a value in (−∞,∞) with the fol- lowing symmetric and bell-shaped density function. f(x) = 1 π[1+(x−µ)2] The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability.
How to compute the expectation of a random variable?
2.3. Expectation and Variance Given a random variable, we often compute the expectation and variance]