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Mean & Variance of a Distribution

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Presentation on theme: "Mean & Variance of a Distribution"β€” Presentation transcript:

1 Mean & Variance of a Distribution

2 Mean & Variance of a Distribution
The mean πœ‡ and variance 𝜎 2 of a random variable X and of its distribution are the theoretical counterparts of the mean π‘₯ and variance s2 of a frequency distribution in the previous section. And serve a similar purpose. Indeed, the mean characterizes the central location and the variance the spread (the variability) of the distribution.

3 Mean & Variance of a Distribution
The mean πœ‡ is defined by

4 Mean & Variance of a Distribution
And the variance 𝜎 2 by

5 Mean & Variance of a Distribution
𝜎 (the positive square root of 𝜎 2 ) is called the standart deviation f X and its distribution. F is the probability function or the density, respectively, in (a) or (b). The mean πœ‡ is also denoted by E(X) and is called the expectation of X because it gives the avarage value of X to be expected in many trials. Quantities such as πœ‡ and 𝜎 2 that measure certain properties of a distribution are called parameters. πœ‡ and 𝜎 2 are the two most important ones.

6 Mean & Variance of a Distribution
From (2) we see that Expect for a discrete β€˜distribution’ with only one possible value, so that 𝜎 2 = 0. We assume that πœ‡ and 𝜎 2 exist (are finite), as is the case for practically all disributions that useful in application.

7 Mean And Variance The random variable X=Number of heads in a single toss of a fair coin has the possible values X=0 and X=1 with probabilities P(X=0)= 1 2 and P(X=1)= 1 2 . Form (1a) we thus obtain the mean πœ‡ = 0 * * = 1 2 , and (2a) yields the variance

8 Uniform Distribution. Variance Measures Spread
The distribution with the density And f=0 otherwise is called the uniform distribution on the interval π‘Ž < x < b.

9 Uniform Distribution. Variance Measures Spread
From (1b) we find that πœ‡ = (a+b)/2 and (2b) yields the variance

10 Uniform Distribution. Variance Measures Spread
Figures illustrates that the spread is large if and only if 𝜎 2 large

11 Uniform Distribution. Variance Measures Spread

12 Mean of a Symmetric Distribution
We can obtain the mean πœ‡ without calculation if a distribution is symmetric. Indeed, you may prove If a distribution is symmetric with respect to x=c, that is, f(c-x) = f(c+x), then πœ‡ = c

13 Transformation of Mean and Variance
(a) If a random variable X has mean πœ‡ and variance 𝜎 2 , then the random variable Has the mean πœ‡* and variance 𝜎 βˆ—2 , where

14 Transformation of Mean and Variance
(b) In particular, the standardized random variable Z corresponding to X, given by Has the mean 0 and variance 1.

15 Proof We prove (5) for a continuous distribution.
To a small interval 𝐼 of length Ξ”x on the x-axis there corresponds the probability f(x)Ξ”x [approximetely; the area of a rectangle of base Ξ”x and height f(x)]. Then the probability f(x)Ξ”x must equal that for the corresponding interval on the x*-axis, that is, f*(x*)Ξ”x* , where f* is the density of X* and Ξ”x* is the length of the interval on the x*-axis corresponding to 𝐼.

16 Proof Hence for differentials we have f*(x*)𝑑x* = f(x)dx.
Also x* = a1 + a2 x by (4) so that (1b) applied to X* gives

17 Proof On the right the first integral rquals 1 by (10) in the previous section. The second integral is πœ‡. This proves (5) for πœ‡*. It implies From this and (2) applied to X*, again using f*(x*)𝑑x* = f(x)dx, we obtain the second formula in (5),

18 Proof For a discrete distrubution the proof of (5) is similar.
Choosing π‘Ž 1 =-πœ‡ / 𝜎 and π‘Ž 2 =-1 / 𝜎 we obtain (6) form (4), writing X* = Z For these π‘Ž 1 , π‘Ž 2 formula (5) gives πœ‡*=0 and 𝜎*2 = 1, as claimed in (b).

19 Expectation, Moments Recall that (1) defines the expectation (the mean) of X, the value of X to be expected on the avarage, written πœ‡ = E(X). More generally, if g(x) is nonconstant and continuous for all x, then g(x) is a random variable. Hence its mathematical expectation or briefly, its expectation E(g(X)) is the value of g(X) to be the avarage, defined by,

20 Expectation, Moments In the first formula f is the probability function of the discrete random variable X. In the second formula, f is the density of the continuous random variable X. Important special cases are the kth moment of X (where k= 1,2,…)

21 Expectation, Moments And the kth central moment of X (k=1,2,…)
This includes the first moment, the mean of X

22 Expectation, Moments It also includes the second central moment, the variance of X For later use you may prove


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