A box contains 5 chips marked 1,2,3,4, and 5. One chip is drawn at random, the...
A box contains 5 chips marked 1,2,3,4, and 5. One chip is drawn at random, the number on it is noted, and the chip is replaced. The process is repeated with another chip. Let X1,X2, and X3 the outcomes of the three draws which can be viewed as a random sample of size 3 from a uniform distribution on integers. a [10 points] What is population from which these random samples are drawn? Find the mean (µ) and variance of this population (σ2)?
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A box contains 5 chips marked 1,2,3,4, and 5. One chip is drawn
at random, the...
Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls...
Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers. a) What is population from which these random samples are drawn? Find the mean (u) and variance of this population (62)? Show your calculations and results.
2. A box contains tickets marked 1,2,3, n. A ticket is drawn at random from the...
2. A box contains tickets marked 1,2,3, n. A ticket is drawn at random from the box. Then this ticket is replaced in the box and a second ticket is drawn at random. Find the probability the second number drawn is smaller than the first. (Recall 1 + 2 + 3 + + k-en! i k(k+1) ).
A box contains seven chips, each of which is numbered (one number on each chip). The...
A box contains seven chips, each of which is numbered (one number on each chip). The number 1 appears on one chip. The number 4 appears on one chip. The number 2 appears on three chips. The number 3 appears on two chips. Two chips are to be randomly sampled from the box without replacement. Let X be the sum of the numbers on the two chips to be sampled. (a) Write out all of the possible outcomes for this...
1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8...
1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8 are marked $2 each and 2 are $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation. Let f(z) = 2z, o < 1, zero elsewhere, be the pdf of X. (a) Compute E(1/X). (b) Find the odf and the pdf of Y...
If a null hypothesis is rejected at a significance level of 1%, then we should say...
If a null hypothesis is rejected at a significance level of 1%,
then we should say that it was rejected at 1%. Reporting that the
null was also rejected at the 5% level of significance is
unnecessary and unwise.
True
False
The p-value equals alpha, the level of significance of the
hypothesis test.
True
False
THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING
INFORMATION:
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with...
Question 1. The random variable X has mean µ and variance o?. Three independent observa- tions...
Question 1. The random variable X has mean µ and variance o?. Three independent observa- tions are drawn, x1, x2, x3. Consider the following estimators of µ: 71) 1.3.x1 – 0.2x2 7(2) 0.8x1 + 0.2x3 (3) ax1 +bx3 7(4) X3 2" 7(5) + X2 + 3 3, 3"3 3" 1.4.1 Which value(s) for a and b would make (3) an unbiased estimator of ? 1.4.2 Which value(s) for a and b would minimize the variance of (3)? 1.4.3 Which value(s)...
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
5. Three boxes are numbered 1, 2 and 3. For k 1, 2, 3, box k contains k blue marbles and 5 - k red marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replace...
5. Three boxes are numbered 1, 2 and 3. For k 1, 2, 3, box k contains k blue marbles and 5 - k red marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replacement. If the probability of selecting box k is proportional to k, then the probability that two marbles drawn have different colours is 6. Two balls are.dropped in such a way that each ball is equally likely to...
Only Questions 4,5 and 6 a=5 Problem 1. Let (X1, ...., Xn) be an i.i.d random...
Only Questions 4,5 and 6
a=5
Problem 1. Let (X1, ...., Xn) be an i.i.d random sample with X; ~ U[0, 2a), and (Y1, ..., Yn) be an i.i.d random sample with Y; ~ Exp ( 1. Find E[X;], E[X3], E[Y/] and E[Y;?). 2. Notwithstanding the actual distributions of the random samples, suppose the modeller believes that they are i.i.d draws from a U (0, 2a distribution. Find the (simple) method of moments estimator â. 3. Let n = 1000....
, Samples In 30) drawn from a uniform distribution la Minitab was used to generate the...
, Samples In 30) drawn from a uniform distribution la Minitab was used to generate the samples. es 300, b 500) Variables 15 Observations Variable TypeFormValues Missing Sample 1 Quantitative Sample 2 Quantitative Numeric Sample 3 Quantitative Numeric Sample 4 Quantitative Sample 5 ive Sample 6 Quantitative Sample 7 Quantitative Observations Sample 8 Quantitative Numeric Sample 9 Quantitative Sample 10 Quantitative Sample 11 Quantitative Sample 12 Quantitative Sample 13 Quantitative Sample 14 Quantitative Sample 15 Quantitative Numeric Numeric Variable Numeric...
Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers. a) What is population from which these random samples are drawn? Find the mean (u) and variance of this population (62)? Show your calculations and results.
2. A box contains tickets marked 1,2,3, n. A ticket is drawn at random from the box. Then this ticket is replaced in the box and a second ticket is drawn at random. Find the probability the second number drawn is smaller than the first. (Recall 1 + 2 + 3 + + k-en! i k(k+1) ).
A box contains seven chips, each of which is numbered (one number on each chip). The number 1 appears on one chip. The number 4 appears on one chip. The number 2 appears on three chips. The number 3 appears on two chips. Two chips are to be randomly sampled from the box without replacement. Let X be the sum of the numbers on the two chips to be sampled. (a) Write out all of the possible outcomes for this...
1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8 are marked $2 each and 2 are $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation. Let f(z) = 2z, o < 1, zero elsewhere, be the pdf of X. (a) Compute E(1/X). (b) Find the odf and the pdf of Y...
If a null hypothesis is rejected at a significance level of 1%,
then we should say that it was rejected at 1%. Reporting that the
null was also rejected at the 5% level of significance is
unnecessary and unwise.
True
False
The p-value equals alpha, the level of significance of the
hypothesis test.
True
False
THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING
INFORMATION:
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with...
Question 1. The random variable X has mean µ and variance o?. Three independent observa- tions are drawn, x1, x2, x3. Consider the following estimators of µ: 71) 1.3.x1 – 0.2x2 7(2) 0.8x1 + 0.2x3 (3) ax1 +bx3 7(4) X3 2" 7(5) + X2 + 3 3, 3"3 3" 1.4.1 Which value(s) for a and b would make (3) an unbiased estimator of ? 1.4.2 Which value(s) for a and b would minimize the variance of (3)? 1.4.3 Which value(s)...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
5. Three boxes are numbered 1, 2 and 3. For k 1, 2, 3, box k contains k blue marbles and 5 - k red marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replacement. If the probability of selecting box k is proportional to k, then the probability that two marbles drawn have different colours is 6. Two balls are.dropped in such a way that each ball is equally likely to...
Only Questions 4,5 and 6
a=5
Problem 1. Let (X1, ...., Xn) be an i.i.d random sample with X; ~ U[0, 2a), and (Y1, ..., Yn) be an i.i.d random sample with Y; ~ Exp ( 1. Find E[X;], E[X3], E[Y/] and E[Y;?). 2. Notwithstanding the actual distributions of the random samples, suppose the modeller believes that they are i.i.d draws from a U (0, 2a distribution. Find the (simple) method of moments estimator â. 3. Let n = 1000....
, Samples In 30) drawn from a uniform distribution la Minitab was used to generate the samples. es 300, b 500) Variables 15 Observations Variable TypeFormValues Missing Sample 1 Quantitative Sample 2 Quantitative Numeric Sample 3 Quantitative Numeric Sample 4 Quantitative Sample 5 ive Sample 6 Quantitative Sample 7 Quantitative Observations Sample 8 Quantitative Numeric Sample 9 Quantitative Sample 10 Quantitative Sample 11 Quantitative Sample 12 Quantitative Sample 13 Quantitative Sample 14 Quantitative Sample 15 Quantitative Numeric Numeric Variable Numeric...
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