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WMDhamnekar
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What is the meaning of this proof? What is the meaning of last statement of this proof? How to prove lemma (7.1)? or How to answer problem 1 given below?
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Dhamnekar Winod said:View attachment 11392
View attachment 11393
What is the meaning of this proof? What is the meaning of last statement of this proof? How to prove lemma (7.1)? or How to answer problem 1 given below?
But, how to use this information to solve the given problem 1 or lemma 7.1?Klaas van Aarsen said:If $x>0$ then the inequality in (1.9) must be true, because the left side is then slightly less than n(x), and the right side is slightly more than n(x).
Let's consider the derivatives of the expressions in (1.8).
$$\frac d{dx}(1-\Re(x)) = -\Re'(x) = -n(x)$$
Let's first find $n'(x)$.
We have:
$$n'(x)=\frac d{dx} \frac 1{\sqrt{2\pi}} e^{-\frac 12x^2} = \frac 1{\sqrt{2\pi}} e^{-\frac 12x^2} \cdot -x =-xn(x)$$
Then we have for instance:
$$\frac d{dx}(x^{-1}n(x)) = -x^{-2}n(x) + x^{-1}n'(x) = -x^{-2}n(x)+x^{-1}\cdot -x n(x) = -(1+x^{-2})n(x)$$
So we see that the derivatives of the expressions in (1.8) are indeed the negatives of the expressions in (1.9).
We have that $1-\Re(x)$ is in between 2 expressions, so its integration must also be between the integrations of the those 2 expressions.
Qed.
To prove the more general formula, we need to repeat these steps for the additional terms.
Write (7.1) in the same form as (1.8) with the series on the left and also on the right.Dhamnekar Winod said:But, how to use this information to solve the given problem 1 or lemma 7.1?
Thanks for your answer. But sorry for not understanding it. I want to know your way of answering this problem.Klaas van Aarsen said:Write (7.1) in the same form as (1.8) with the series on the left and also on the right.
Take the derivatives to find an expression that is in the same form as (1.9).
Then the proof follows in the same fashion.
The normal distribution tail inequality is a mathematical concept that states that the probability of a random variable falling outside a certain range, defined by a specified number of standard deviations from the mean, is very small. In other words, the probability of a value being extremely large or extremely small is very low in a normal distribution.
The formula for the normal distribution tail inequality is P(|X| > x) < 2e-x2/2, where X is a random variable following a normal distribution with mean 0 and standard deviation 1.
The proof for the normal distribution tail inequality for large x involves using properties of the standard normal distribution and the concept of integration. By integrating the probability density function of the standard normal distribution, we can show that the probability of a value being larger than a certain threshold decreases exponentially as the threshold increases.
The normal distribution tail inequality is important because it helps us understand the behavior of random variables that follow a normal distribution. It allows us to make predictions about the likelihood of extreme values occurring, which is useful in many fields such as statistics, finance, and engineering.
While the normal distribution tail inequality specifically applies to the normal distribution, similar concepts and inequalities can be applied to other distributions. For example, the Chebyshev inequality can be used to make statements about the probability of values falling within a certain range for any distribution, not just the normal distribution.