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## Statistics and Probability

### Fundamentals of Probability - Random Variables

#### Q.01

'Since the relative frequency R follows the same distribution as the sample ratio, R is approximately normal with N(1/6, 1/6(1-1/6) * 1/n) i.e., N(1/6, 5/36n). Therefore, if we define Z=(R-1/6)/(1/6 * sqrt(5/n)), Z is approximately distributed as N(0,1). Hence, \\[P(|R-1/6| ≤ 1/60)=P(1/6 sqrt(5/n)|Z| ≤ 1/60) \\]\n\\[\egin{array}{l}=P(|Z| ≤ 1/10 sqrt(n/5))=P(-1/10 sqrt(n/5) ≤ Z ≤ 1/10 sqrt(n/5))\\end{array} \\]\nTherefore, the required values are when n=500\n\\[P(-1 ≤ Z ≤ 1)=2 p(1)=2 * 0.3413=0.6826 \\]n=2000\n\\[P(-2 ≤ Z ≤ 2)=2 p(2)=2 * 0.4772=0.9544 \\]n=4500\n\\[P(-3 ≤ Z ≤ 3)=2 p(3)=2 * 0.49865=0.9973'

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'Please create a table describing the probability distribution of the random variable X.'

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'Sampling from a population by selecting one item at a time with replacement each time is known as sampling with replacement. In contrast, continuing to sample without replacement after selecting is known as sampling without replacement. Randomly selecting a sample of size n from a population, and assigning the values of the variables in those n elements as X₁, X₂, ..., Xₙ. When sampling with replacement, it can be considered as a repeated experiment of randomly selecting a sample of size 1 n times. Therefore, X₁, X₂, ..., Xₙ are independent random variables that each follow the population distribution.'

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'Find the corresponding point for z = 5.93 based on the given table.'

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'What is the probability distribution of the number of winning tickets drawn before drawing two losing tickets in a lottery where there are n (n is an integer greater than or equal to 3) tickets, of which 582 are losing tickets? '

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'The support rate of political party A among eligible voters nationwide is 32%. Let the random variable Xk correspond to the value of 1 if the k-th randomly selected person supports party A, and 0 if they do not, out of 100 randomly sampled voters. Find the expected value E(X̄) and standard deviation σ(X̄) of the sample mean X̄.'

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'What is the probability p_{n+1} that two particles are at the same point after (n+1) seconds?'

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'When two random variables X and Y, the product XY is also a random variable, and X and Y are independent of each other, the following theorem holds true. E(XY) = E(X)E(Y)。'

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'Use symbols to express the probability that the random variable X takes a value greater than or equal to a and less than or equal to b.'

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'Example 67 Sample Proportion and Normal Distribution'

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'Chapter 2 Statistical Inference 8. Random Variables and Probability Distributions 9. Transformation of Random Variables 10. Sum of Random Variables and Expectation 11. Binomial Distribution 12. Normal Distribution 13. Population and Sample, Sample Mean and Its Distribution 14. Estimation 15. Hypothesis Testing'

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'When throwing a dice n times, let R be the relative frequency of getting a 1. Find the value of P(|R-\\frac{1}{6}| \\leqq \\frac{1}{60}) for n=500, 2000, and 4500.'

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'Transformation of random variables\nX is a random variable, a and b are constants.\nWhen Y=aX+b\nE(Y)=aE(X)+b\nV(Y)=a^{2}V(X)\\sigma(Y)=|a|\\sigma(X)'

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'Suppose the joint distribution of two random variables X and Y is given as follows:'

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'For two random variables X, Y, if X and Y are independent of each other, then V(X+Y) = V(X) + V(Y).'

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'Choose one of the two definitions of the negative binomial distribution, and calculate the probability of the number of trials X until event A occurs k times, or the number of failures Y.'

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'When taking a random sample of size 100 from a population following a normal distribution with a population mean of 58 and a population standard deviation of 12, calculate the following probabilities.'

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"Let the probability of event A occurring in one trial be p. In a repeated experiment of n trials, the probability of A occurring exactly r times is ${}_n C_r p^r q^{n-r}$, where q=1-p. In mathematical analysis A, we studied about probabilities in repeated experiments. In a repeated experiment, the number of times an event occurs is denoted by X, making X a random variable. Let's delve into this."

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'Since the sample size n is 900, the 95% confidence interval for the population mean m is X - 1.96*(9.8 / sqrt(900)) <= m <= X + 1.96*(9.8 / sqrt(900))'

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'Calculate the 95% confidence interval for the population mean m.'

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'The range of the random variable X is 0 ≤ X ≤ 1, and its probability density function is f(x)=a(2-x). Where a is a positive constant.\n(1) Find the value of a.\n(2) Find the expected value E(X) and variance V(X) of the random variable X.'

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'In the confidence interval A <= m <= B, the 99% confidence interval E <= m <= F for the population mean m obtained from the same sample will have what effect on the width of the range compared to A <= m <= B? Please select from the following options:'

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'When throwing two dice simultaneously, calculate the probability distribution of the following random variable X. However, for (2), if the two dice show the same number, take that number as X. (1) The difference between the numbers on the two dice is X. (2) The minimum number rolled is X. There are 36 possible outcomes when throwing two dice simultaneously.'

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'When two dice are rolled simultaneously, let the smaller number be X. Find the following: Take the number itself if the same number appears.'

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'Throw three dice of large, medium, and small sizes simultaneously. Use the numbers on the large, medium, and small dice as the hundreds, tens, and units digits, respectively, to create a three-digit integer. Find the following expected values:\n(1) Expected value of the sum of the digits\n(2) Expected value of the three-digit integer'

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'Let n be a natural number greater than or equal to 8. From 1, 2, ..., n, randomly select 6 different numbers and arrange them in ascending order as X_{1}<X_{2}<X_{3}<X_{4}<X_{5}<X_{6}.\n(1) Find the probability p_{n} that X_3=5.\n(2) Find the natural number n that maximizes p_n.'

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'Point P is initially at the origin O on the number line, and each time a die is thrown, if an even number appears, move 3 units in the positive direction, and if an odd number appears, move 2 units in the negative direction. When the die is thrown 10 times, the probability that point P is at the origin O is A. Also, when the die is thrown 10 times, the probability that the coordinate of point P is less than or equal to 19 is B.'

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'The probability of getting heads exactly 5 times out of 6 tosses is ${}_6 C_5 (\\frac{1}{2})^5 (1-\\frac{1}{2})^{6-5}=6 \\times (\\frac{1}{2})^5 \\times \\frac{1}{2}=\\frac{6}{64}$. The probability of getting heads all 6 times out of 6 tosses is $(\\frac{1}{2})^6=\\frac{1}{64}$. The sum of these probabilities is $\\frac{15}{64}+\\frac{6}{64}+\\frac{1}{64}=\\frac{22}{64}=\\frac{11}{32}$.'

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"One would like to investigate the effectiveness of a drug administered for a certain disease. Assuming that the ratio of those judged to be effective after administering the drug is 33% to 63%. Randomly selecting n individuals from patients with the disease, defining a random variable X_i as 1 if the drug's effectiveness is observed in the i-th patient, and 0 otherwise.\n(1) Determine the mean and variance of the sample mean \ \\overline{X}=\\frac{1}{n} \\sum_{i=1}^{n} X_{i} \.\n(2) Selecting 400 individuals randomly from patients with the disease, 320 individuals were observed to have the drug's effectiveness. Determine the 95% confidence interval for the population proportion, rounding to the third decimal place. Assume the sample size of 400 is sufficiently large. [Kyushu University]"

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'The approval rating of party A among voters in a certain city is 64%. When 100 people are randomly selected from the voters in this city, let the random variable that assigns a value of 1 if the k-th person selected supports party A and 0 if not be denoted as X_k.'

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'When extracting samples of size 2 from the population {A, B, C, D}, list all possible samples in each case. (1) With replacement (2) Without replacement - [1] drawn consecutively [2] drawn simultaneously (3) 1!'

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'In a bag, there are 1 white ball, 2 red balls, and 3 blue balls. When 2 balls are drawn from the bag without replacement, let the number of red balls drawn be X and the number of blue balls drawn be Y. Find the joint distribution of X and Y.'

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'Given the probability distributions of random variables X and Y in the following table, find Var(3X+2Y) and Var(6X-4Y). Assuming X and Y are independent.'

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'Considering all students at P University as the population, with a mother proportion of 0.2 for individuals from A prefecture, and a random sample size of 400, the random variable X follows a binomial distribution B(400,0.2). Therefore, the expected value E(X) and standard deviation σ(X) of X are E(X)=400⋅0.2=80 σ(𝐗)=√(400⋅0.2⋅(1−0.2))=√(8^2)=8'

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'Among all students of P University, 20% are from A prefecture. Let X be the number of A prefecture residents among 400 students randomly sampled from P University. Find the expected value and standard deviation of X.'

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'When a six-sided die is rolled 360 times, let X be the number of times 6 appears. Determine the probability of X falling within the following ranges. Assume that √2 = 1.41.\n(1) 50 ≤ X ≤ 60\n(2) |X/360 - 1/6| ≤ 0.05'

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'Assume a student population where the proportion of students who have never read a book is 0.5. In this case, let X be the random variable representing the number of students who have never read a book in a random sample of 100 students. What distribution does X follow? Also, what is the mean (expected value) and standard deviation of X.'

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"Want to investigate the effectiveness of a drug administered for a certain disease. Let's assume that the ratio of patients judged to have an effect after receiving the drug is p. Select n patients with the disease randomly, if the drug effect is observed in the i-th patient, then it's 1, otherwise 0, defining the random variable Xi."

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'It is known that the ratio of newborn boys to girls in city A is equal. In a particular year, when n individuals are randomly sampled from newborns in city A, let Xk be the random variable that assigns a value of 1 if the k-th newborn is a boy and 0 if a girl.'

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'For a random variable X following a normal distribution N(m, σ^2), when Z=(X-3)/5 follows the standard normal distribution N(0,1), find the values of m and σ.'

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'Consider a game where a dice is rolled, earning 0 points for rolling 1 or 2, 1 point for rolling 3, 4, or 5, and 100 points for rolling a 6. Let X be the remainder when the total score from 80 rolls is divided by 100. Find the probability that X is less than or equal to 46. Given that √5 = 2.24.'

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'The table on the right summarizes the scores of two math and English tests conducted twice in a small class of 10 students, totaling 100 points.'

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"In the scatter plot on the right, 187,198 math exercises are tests of 100 points in Chinese characters and English words in a class of 30 people. （1）Based on this scatter plot, determine whether there is a correlation between the scores of Chinese characters and English words. If there is a correlation, state whether it is positive or negative. （2）Based on this scatter plot, create a frequency distribution table for English words. However, the classes are '40 or more but less than 50', ..., '90 or more but less than 100'. The points on the scatter plot are distributed upwards and to the right as a whole. Count based on the horizontal lines incrementing by 10 points of English words."

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"Ski jumping is a sport where athletes compete based on the distance of their jump and the beauty of their mid-air posture. Competitors slide down a slope and then launch themselves into the air from the edge of the slope. The distance of the jump (measured in meters) determines score X, while the mid-air posture determines score Y. Let's consider 58 jumps in a specific competition.\n(1) Based on the three scatter plots in Figure 1, select the correct statements:\n1. There is a positive correlation between X and Y.\n2. The jump with the highest speed V also has the highest X.\n3. The jump with the highest speed V also has the highest Y.\n4. The jump with the lowest Y does not necessarily have the lowest X.\n5. All jumps with X greater than or equal to 80 have a speed V of 93 or higher.\n6. There are no jumps with Y greater than or equal to 55 and V greater than or equal to 94."

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'Example 1. In an experiment of rolling a die once, event A: getting an odd number, event B: getting a number 4 or higher, then A={1,3,5}, B={4,5,6}, so A ∩ B={5} is odd and 4 or higher. A ∪ B={1,3,4,5,6} is odd or 4 or higher.'