If x is the mean of a distribution, then E f (x - x)
is equal to
Answers
Answer:
When introducing the topic of random variables, we noted that the two types — discrete and continuous — require different approaches.
In the module Discrete probability distributions , the definition of the mean for a discrete random variable is given as follows: The mean μXμX of a discrete random variable XX with probability function pX(x)pX(x) is
μX=E(X)=∑xpX(x),μX=E(X)=∑xpX(x),
where the sum is taken over all values xx for which pX(x)>0pX(x)>0.
The equivalent quantity for a continuous random variable, not surprisingly, involves an integral rather than a sum. The mean μXμX of a continuous random variable XX with probability density function fX(x)fX(x) is
μX=E(X)=∫∞−∞xfX(x)dx.μX=E(X)=∫−∞∞xfX(x)dx.
By analogy with the discrete case, we may, and often do, restrict the integral to points where fX(x)>0fX(x)>0.
Several of the points made when the mean was introduced for discrete random variables apply to the case of continuous random variables, with appropriate modification.
Recall that mean is a measure of 'central location' of a random variable. It is the weighted average of the values that XX can take, with weights provided by the probability density function. The mean is also sometimes called the 'expected value' or 'expectation' of XX and denoted by E(X)E(X). In visual terms, looking at a pdf, to locate the mean you need to work out where the pivot should be placed to make the pdf balance on the xx-axis, imagining that the pdf is a thin plate of uniform material, with height fX(x)fX(x) at xx.
An important consequence of this is that the mean of any symmetric random variable (continuous or discrete) is always on the axis of symmetry of the distribution; for a continuous random variable, this means the axis of symmetry of the pdf.
Exercise 3
Two triangular pdfs are shown in figure 9.

Figure 9: The probability density functions of two continuous random variables.
Each of the pdfs is equal to zero for x<0x<0 and x>10x>10, and the xx-values of the apex and the boundaries of the shaded region are labelled on the xx-axis in figure 9.
For each of these pdfs separately:
Write down a formula (involving cases) for the pdf.
Guess the value of the mean. Then calculate it to assess the accuracy of your guess.
Guess the probability that the corresponding random variable lies between the limits of the shaded region. Then calculate the probability to check your guess.
The module Discrete probability distributions gives formulas for the mean and variance of a linear transformation of a discrete random variable. In this module, we will prove that the same formulas apply for continuous random variables.
Theorem
Let XX be a continuous random variable with mean μXμX. Then
<
Step-by-step explanation:
Plz follow me plz
plz mark me as brainliest answer