What is the intuition behind the binomial variance

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»Definition of the expected value
»Variance and standard deviation
“Examples
" Remarks

Definition of the expected value

The expected value \ (E (X) \), often also \ (\ lambda \) or \ (\ mu \), is colloquially the value that is most likely to occur. More precisely, it only marks one area, because we shall see, for example, that \ (\ mu = 3.5 \) occurs when the dice is thrown.

For a discrete random variable with event space \ (\ Omega = {\ omega_1, \ dots, \ omega_n} \) the expected value applies
\ begin {align *}
E (X) & = \ sum_ {i = 1} ^ n \ omega_i \ cdot P (X = \ omega_i) = \
& \ omega_1 \ cdot P (X = \ omega_1) + \ dots \ omega_n \ cdot P (X = \ omega_n).
\ end {align *}
For a discrete random variable with a countably infinite event space \ (\ Omega = {\ omega_1, \ dots} \) the expected value applies
\ begin {align *}
E (X) & = \ sum_ {i = 1} ^ \ infty \ omega_i \ cdot P (X = \ omega_i) = \
& \ omega_1 \ cdot P (X = \ omega_1) + \ omega_2 \ cdot P (X = \ omega_2) + \ dots.
\ end {align *}
For a continuous random variable with event space \ (\ mathbb {R} \) the expected value applies
\ begin {align *}
E (X) & = \ int _ {- \ infty} ^ \ infty x \ cdot f (x) dx.
\ end {align *}
Variance and standard deviation

Similar to statistics, the variance measures the dispersion around the expected value; we quote ourselves from statistics, "More precisely, the variance measures the mean deviation from the arithmetic mean." It weights values ​​close to the expected value less strongly than values ​​further away due to the squaring.

Examples

The dice roll: How is the expected value of the dice roll calculated, the result of which we have already anticipated? We have \ (\ Omega = {1, 2, \ dots, 6} \) and if it is a fair Laplace cube, all events are equally likely with probability \ (\ frac {1} {6} \) . So follows:

\ begin {align *}
E (X) & = \ sum_ {i = 1} ^ 6 \ omega_i \ cdot P (X = \ omega_i) = \
& = 1 \ times P (X = 1) +2 \ times P (X = 2) + \ dots +6 \ times P (X = 6) = \
& = 1 \ cdot \ frac {1} {6} +2 \ cdot \ frac {1} {6} + \ dots +6 \ cdot \ frac {1} {6} = \
& = 3.5: = \ mu.
\ end {align *}

 

Similarly, we strictly follow the formula for the variance
\ begin {align *}
Var (X) & = \ sum_ {i = 1} ^ 6 (\ omega_i- \ mu) ^ 2 \ cdot P (X = \ omega_i) = \
& = (1-3.5) ^ 2 \ times P (X = 1) + (2-3.5) ^ 2 \ times P (X = 2) + \ dots + (6-3.5) ^ 2 \ cdot P (X = 6) = \
& = 2.5 ^ 2 \ cdot \ frac {1} {6} + 1.5 ^ 2 \ cdot \ frac {1} {6} + \ dots + 2.5 ^ 2 \ cdot \ frac {1} { 6} = \
& =2,91667.
\ end {align *}

 

Totals: What are the expected value and the variance of the totals of the two-time dice roll? We have already created the following table beforehand.

 

\ (x \)234567
\ (P (X = x) \)\ (\ frac {1} {36} \)\ (\ frac {2} {36} \)\ (\ frac {3} {36} \)\ (\ frac {4} {36} \)\ (\ frac {5} {36} \)\ (\ frac {6} {36} \)
\ (x \)89101112 
\ (P (X = x) \)\ (\ frac {5} {36} \)\ (\ frac {4} {36} \)\ (\ frac {3} {36} \)\ (\ frac {2} {36} \)\ (\ frac {1} {36} \) 

 

We apply our formulas again:
\ begin {align *}
E (X) & = \ frac {1} {36} \ cdot 2 + \ frac {2} {36} \ cdot 3+ \ dots + \ frac {1} {36} \ cdot 12 = \
& = 7.
\ end {align *}
Similarly, we get for the variance
\ begin {align *}
Var (x) & = \ frac {1} {36} \ cdot (2-7) ^ 2 + \ frac {2} {36} \ cdot (3-7) ^ 2 + \ dots + \ frac {1} {36} \ cdot (12-7) ^ 2 = \
& = \ frac {35} {6} \ approx 5.83 \
\ Rightarrow \ sigma & = \ sqrt {5.83} = 2.42.
\ end {align *}

 

Remarks

In contrast to the continuous random variables, this chapter is a little helpless with regard to examples, since we have hardly considered any examples. In the respective chapters of the continuous random variables, however, there is always information on the expected value and variance.

The formula of the expected value is very similar to that of the arithmetic mean. They also calculate very similar things, the average value of a data set and the expected average value of a random attempt. Exactly defined calculated \ (Var (X) = E (X- \ mu) ^ 2) \).

Often, instead of \ (\ omega_i \), based on the functions and the continuous random variable, \ (x_i \) and \ (f (x_i) \) are used, for example
\ begin {align *}
E (X) & = \ sum_ {i = 1} ^ n x_i \ cdot P (X = x_i) = \ sum_ {i = 1} ^ n x_i \ cdot f (x_i).
\ end {align *}

The standard deviation is often preferred to the variance in mathematics. If our random variable \ (X \) has a unit, for example \ (cm \), the variance has the unit \ (cm ^ 2 \), but the standard deviation, "due to the extraction of the root", \ (cm \). So it has the same unit as the expected value. There is a rule of thumb in statistics which says that in the area \ (E (X) \ pm \ sigma \) "most things happen" and in the area \ (E (X) \ pm 2 \ sigma \) "almost everything". In the example of the double roll of the dice, we get the area \ ([\ mu - \ sigma, \ mu + \ sigma] = [4.58; 9.42] \) where almost all dice are thrown and in the area \ ([\ mu - 2 \ sigma, \ mu + 2 \ sigma] = [2,16; 11,48] \) then there are actually almost all possible results. You can find more about this in the descriptive statistics.