For a population following the normal distribution $\mathrm { N } \left( 20,5 ^ { 2 } \right)$, a sample of size 16 is randomly extracted and the sample mean is denoted by $\bar { X }$. What is the value of $\mathrm { E } ( \bar { X } ) + \sigma ( \bar { X } )$? [3 points]
(1) $\frac { 83 } { 4 }$
(2) $\frac { 85 } { 4 }$
(3) $\frac { 87 } { 4 }$
(4) $\frac { 89 } { 4 }$
(5) $\frac { 91 } { 4 }$

The random variable $X$ follows a normal distribution with mean 8 and standard deviation 3, and the random variable $Y$ follows a normal distribution with mean $m$ and standard deviation $\sigma$. The two random variables $X$ and $Y$ satisfy $$\mathrm { P } ( 4 \leq X \leq 8 ) + \mathrm { P } ( Y \geq 8 ) = \frac { 1 } { 2 }$$ Find the value of $\mathrm { P } \left( Y \leq 8 + \frac { 2 \sigma } { 3 } \right)$ using the standard normal distribution table on the right.

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

[3 points]
(1) 0.8351
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938

A random variable $X$ follows a normal distribution with mean 8 and standard deviation 3, and a random variable $Y$ follows a normal distribution with mean $m$ and standard deviation $\sigma$. If the two random variables $X$ and $Y$ satisfy $$\mathrm { P } ( 4 \leq X \leq 8 ) + \mathrm { P } ( Y \geq 8 ) = \frac { 1 } { 2 }$$ find the value of $\mathrm { P } \left( Y \leq 8 + \frac { 2 \sigma } { 3 } \right)$ using the standard normal distribution table below.

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

[4 points]
(1) 0.8351
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938

For a positive number $t$, the random variable $X$ follows a normal distribution $\mathrm{N}(1, t^2)$. $$\mathrm{P}(X \leq 5t) \geq \frac{1}{2}$$ For all positive numbers $t$ satisfying this condition, find the maximum value of $\mathrm{P}(t^2 - t + 1 \leq X \leq t^2 + t + 1)$ using the standard normal distribution table below, and let this value be $k$. Find the value of $1000 \times k$. [4 points]

$z$	$\mathrm{P}(0 \leq Z \leq z)$
0.6	0.226
0.8	0.288
1.0	0.341
1.2	0.385
1.4	0.419

A random variable $X$ follows a normal distribution $\mathrm{N}\left(m_{1}, \sigma_{1}^{2}\right)$ and a random variable $Y$ follows a normal distribution $\mathrm{N}\left(m_{2}, \sigma_{2}^{2}\right)$, satisfying the following conditions. For all real numbers $x$, $\mathrm{P}(X \leq x) = \mathrm{P}(X \geq 40 - x)$ and $\mathrm{P}(Y \leq x) = \mathrm{P}(X \leq x + 10)$. When $\mathrm{P}(15 \leq X \leq 20) + \mathrm{P}(15 \leq Y \leq 20) = 0.4772$ using the standard normal distribution table below, what is the value of $m_{1} + \sigma_{2}$? (Given: $\sigma_{1}$ and $\sigma_{2}$ are positive.) [4 points]

$z$	$\mathrm{P}(0 \leq Z \leq z)$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

For a natural number $a$ not exceeding 6, a trial is performed using one die and one coin.
Roll the die once. If the result is less than or equal to $a$, flip the coin 5 times and record the number of heads. If the result is greater than $a$, flip the coin 3 times and record the number of heads.
This trial is repeated 19200 times, and let $X$ be the number of times the recorded number is 3. When $\mathrm { E } ( X ) = 4800$, find the value of $\mathrm { P } ( X \leq 4800 + 30 a )$ using the standard normal distribution table below, which equals $k$.

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.5	0.191
1.0	0.341
1.5	0.433
2.0	0.477
2.5	0.494
3.0	0.499

Find the value of $1000 \times k$. [4 points]

6. The measurement result of a certain physical quantity follows a normal distribution $N \left( 10 , \sigma ^ { 2 } \right)$. Which of the following conclusions is incorrect? ( )
A. The smaller $\sigma$ is, the greater the probability that the physical quantity falls in $( 9.9,10.1 )$ in a single measurement.
B. The smaller $\sigma$ is, the probability that the physical quantity is greater than 10 in a single measurement is 0.5.
C. The smaller $\sigma$ is, the probability that the physical quantity is less than 9.99 equals the probability that it is greater than 10.01 in a single measurement.
D. The smaller $\sigma$ is, the probability that the physical quantity falls in $( 9.9,10.2 )$ equals the probability that it falls in $( 10,10.3 )$ in a single measurement.
【Answer】D

【Solution】

【Analysis】Use the properties of the normal distribution density curve to judge each option. 【Detailed Solution】For option A, $\sigma ^ { 2 }$ is the variance of the data, so the smaller $\sigma$ is, the more concentrated the data is around $\mu = 10$. Therefore, the probability that the measurement result falls in $( 9.9,10.1 )$ increases, so A is correct.
For option B, by the symmetry of the normal distribution density curve, the probability that the physical quantity is greater than 10 in a single measurement is 0.5, so B is correct.
For option C, by the symmetry of the normal distribution density curve, the probability that the physical quantity is greater than 10.01 equals the probability that it is less than 9.99 in a single measurement, so C is correct.
For option D, since the probability that the physical quantity falls in $( 9.9,10.0 )$ does not equal the probability that it falls in $( 10.2,10.3 )$, the probability that the measurement result falls in $( 9.9,10.2 )$ does not equal the probability that it falls in $( 10,10.3 )$, so D is incorrect. Therefore, the answer is: D.

The measurement result of a certain physical quantity follows a normal distribution $N \left( 10 , \sigma ^ { 2 } \right)$. Which of the following conclusions is incorrect?
A. The smaller $\sigma$ is, the greater the probability that the physical quantity falls in $( 9.9,10.1 )$ in a single measurement.
B. The smaller $\sigma$ is, the probability that the physical quantity is greater than 10 in a single measurement is 0.5.
C. The smaller $\sigma$ is, the probability that the physical quantity is less than 9.99 equals the probability that it is greater than 10.01 in a single measurement.
D. The smaller $\sigma$ is, the probability that the physical quantity falls in $( 9.9,10.2 )$ equals the probability that it falls in $( 10,10.3 )$ in a single measurement.

To understand the per-acre income (in units of 10,000 yuan) after promoting exports, a sample was taken from the planting area. The sample mean of per-acre income after promoting exports is $\bar { x } = 2.1$ , and the sample variance is $s ^ { 2 } = 0.01$ . The historical per-acre income $X$ in the planting area follows a normal distribution $N \left( 1.8 , ~ 0.1 ^ { 2 } \right)$ . Assume that the per-acre income $Y$ after promoting exports follows a normal distribution $N \left( \bar { x } , s ^ { 2 } \right)$ . Then (if a random variable $Z$ follows a normal distribution $N \left( \mu , \sigma ^ { 2 } \right)$ , then $P ( Z < \mu + \sigma ) \approx 0.8413$ )
A. $P ( X > 2 ) > 0.2$
B. $P ( X > 2 ) < 0.5$
C. $P ( Y > 2 ) > 0.5$
D. $P ( Y > 2 ) < 0.8$

Determine the probability that at least 25 hand carts are rented.

Given that in an experiment with 400 plants the value of the random variable $X _ { 400 }$ deviates from the expected value by at most one standard deviation, determine the smallest and largest possible relative frequency of plants that become infested with fungi.

In general, the following inequality holds for a random variable $X$ with expected value $\mu$ and standard deviation $\sigma$ for $k > 0$ :
$$P ( \mu - k \cdot \sigma < X < \mu + k \cdot \sigma ) \geq 1 - \frac { 1 } { k ^ { 2 } }$$
Explain the statement of this inequality for $k = 2$.

At the beginning of 2021 in Germany, approximately 320000 cars with purely electric drive and 280000 plug-in hybrids were registered, thus a total of approximately 600000 cars with electric motors. The share of cars with electric motors in the total stock of all cars registered in Germany was around $1.2 \%$. Determine the number of cars that must be randomly selected from this total stock so that with a probability of more than $97 \%$ at least one car with purely electric drive is among them.

The beverage manufacturer acquires another machine. It is assumed that the filling quantities of all bottles are independent of each other. The continuous random variable $Y$: ``Filling quantity of a randomly selected bottle filled by this machine'' is assumed to be normally distributed with expected value $\mu = 331$ [ml] and standard deviation $\sigma = 1.34 [ \mathrm { ml } ]$. A filling with at most 327 ml is referred to in the following as a severe underfilling.
(1) Determine the probability that a randomly selected bottle is a severe underfilling.
Give your result rounded to five decimal places.
[Check solution with four decimal places: 0.0014.]
(2) Determine the expected number of severe underfilled bottles in a sample of 1500 bottles.
(3) Determine the probability that a sample of 750 bottles contains more than two severe underfilled bottles.
(4) The beverage manufacturer changes the machine parameters to $\mu _ { \text {new } } = 330 [ \mathrm { ml } ]$ and $\sigma _ { \text {new } } = 1.00 [ \mathrm { ml } ]$.
Interpret the changed parameters in the given context.
Assess how the probability that a randomly selected bottle is a severe underfilling changes due to the change in parameters.

Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote $$S_{k} = \sum_{i=1}^{k} U_{i}$$
Deduce Hoeffding's inequality for $S_{k}$: for all $t > 0$, we have $$\mathbb{P}\left(S_{k} \geqslant t\right) \leqslant \exp\left(-\frac{t^{2}}{2k}\right).$$

Deduce that, for every polynomial $P \in \mathbb{R}[X]$, $$\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} P\left(\Lambda_{i,n}\right)\right) = \frac{1}{2\pi} \int_{-2}^{2} P(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$

Deduce a simple equivalent of $1 - \Phi ( x )$ as $x$ tends to $+ \infty$, where $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$ and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

We now assume that the random variables $X_1, \ldots, X_N$ are independent, so that they are $k$-independent for all $k \in \{2, \ldots, N\}$. We now want to establish the following bound: there exist numerical constants $\alpha, \beta > 0$ (independent of $K \geq 1$ and $N$) such that for all $t \geq 0$, $$\mathbb{P}\left(|S_N| \geq t\sqrt{N}\right) \leq \beta \exp\left(-\alpha t^2/K^2\right)$$
I.6.a) Justify that it suffices to consider the case $K = 1$, which we will do in the next three questions.
I.6.b) Let $k$ be the largest even integer in $\{1, \ldots, N\}$ less than or equal to $\frac{2t^2}{e^2}$. Justify that (27) is satisfied if $$e \leq t \leq \frac{e}{\sqrt{2}}\sqrt{N}$$
I.6.c) Under hypothesis (32), prove that we have (31) with $$\beta = 2e, \quad \alpha = e^{-2}$$
I.6.d) Conclude that there exist numerical constants $\alpha, \beta > 0$ such that (31) is verified for all $t \geq 0$.

The lifespan of individuals of a certain animal species has a normal distribution with a mean of 8.8 months and a standard deviation of 3 months.\ a) (1 point) What percentage of individuals of this species exceed 10 months? What percentage of individuals have lived between 7 and 10 months?\ b) (1 point) If 4 specimens are randomly selected, what is the probability that at least one does not exceed 10 months of life?\ c) ( 0.5 points) What value of $c$ is such that the interval ( $8.8 - c , 8.8 + c$ ) includes the lifespan (measured in months) of $98 \%$ of the individuals of this species?

The length of the Pacific sardine (Sardinops sagax) can be considered a random variable with normal distribution with mean 175 mm and standard deviation 25.75 mm.\ a) (1 point) A canning company for this variety of sardines only accepts as quality sardines those with a length greater than 16 cm. What percentage of the sardines caught by a fishing vessel will be of the quality expected by the canning company?\ b) ( 0.5 points) Find a length t $< 175 \mathrm {~mm}$ such that between t and 175 mm there are 18\% of the sardines caught.\ c) (1 point) At sea, sardines are processed in batches of 10. Subsequently, sardines from each batch that are smaller than 15 cm are returned to the sea as they are considered small. What is the probability that in a batch there is at least one sardine returned as small?

10. The figure below is a histogram based on the weights of 100 women (the percentages in the figure represent the relative frequency of each weight interval, where each interval does not include the left endpoint but includes the right endpoint). The mean weight of the 100 women is 55 kg, and the standard deviation is 12.5 kg. Curve N represents a normal distribution with the same mean and standard deviation as the sample values. In this sample, if "overweight" is defined as weight exceeding the sample mean by 2 or more standard deviations (i.e., weight exceeding 80 kg or more), which of the following statements are correct? [Figure]
(1) In curve N (normal distribution), the proportion at 55 kg or above is approximately 50\%.
(2) In curve N (normal distribution), the proportion at 80 kg or above is approximately 2.5\%.
(3) In this sample, the median weight is greater than 55 kg.
(4) In this sample, the first quartile of weight is greater than 45 kg.
(5) In this sample, the proportion of "overweight" (weight exceeding 80 kg or more) is greater than or equal to 5\%.