Question

In: Statistics and Probability

. (1) A man has a 4-sided (tetrahedral) die, with four faces, 1, 2, 3, and...

. (1) A man has a 4-sided (tetrahedral) die, with four faces, 1, 2, 3, and 4. He rolls it 12 times. (a) Given that for each roll of the die, the probability of rolling a 4 is 1/4 and the probability of not doing so is 3/4, compute the following probabilities P(x = x0),where x is the number of times that the man rolls a 4 among the 12 trials. (b) What is the average value of x? What is the standard deviation? (c) Which possible value of x has the highest likelihood of occurring? (2) IQ scores in America, by definition, are normally distributed, with 100 points being the mean IQ and the standard deviation being 10 points. (a) Suppose a person has an IQ of 115. What percentage of the population is expected to have a higher IQ than this person? (b) Now suppose a person has an IQ of 75. What percentage of the population is expected to have a lower IQ than this person? (c) An individual’s IQ was claimed to have jumped from 83 to 212 after a procedure. (An IQ of 212 is not possible, given that most tests only can assign scores of at most 170.) What percentage of the population, were the claim not false, would be between the first score and the second?

Expert Solution

1) the probability of rolling a 4 is p=1/4.

Let X be the random variable which indicates the the number of times that a 4 is rolled in n=12 trials. X has a Binomial distribution with parameters, number of trials n=12 and success probability p=1/4

a) The probability that number of 4 are rolled among 12 trials is

b) The average value of X is the expected value of X. Since X has a Binomial distribution, the expected value of X is

The average value of X is 3

The standard deviation of X is

c) X can take the values from 0 to 12. The different probabilities are

Since P(X=3)=0.2581 is the highest, X=3, 4s out of 12 rolls, has the highest likelihood of occuring.

2) Let Y be the IQ of a randomly selected person. Y has a normal distribution with mean and standard deviation

a) If a person has an IQ of 115, the proportion of the population that is expected to have an IQ higher than 115 is same as the probability that Y>115

The percentage of the population that is expected to have a higher IQ than a person with an IQ of 115 is 6.68%

b) If a person has an IQ of 75, the proportion of the population that is expected to have an IQ lower than 75 is same as the probability that Y<75

The percentage of the population that is expected to have a lower IQ than a person with an IQ of 75 is 0.62%

c) The proportion of the population, were the claim not false, that would be between the first score (83) and the second (212) is same as the probability that 83<Y<212

The percentage of the population, were the claim not false, that would be between the first score (83) and the second (212) is 95.54%

orchestra answered 1 year ago

Suppose we keep rolling a tetrahedral die (with faces marked as 1, 2, 3, 4) till...

Suppose we keep rolling a tetrahedral die (with faces marked as 1, 2, 3, 4) till an even number appears for the first time. (a) Give a precise description of the sample space. (b) Give the probability of each elementary outcome (each element of the sample space). (c) Find the probability of an even number appearing for the first time at the nth roll. (d) Find the probability of an even number appearing for the first time no later than...

Consider a rolling two four-sided dice with faces 1, 2, 3 and 4. (a) Obtain the...

Consider a rolling two four-sided dice with faces 1, 2, 3 and 4. (a) Obtain the pmf of X where X is the sum of the two resultant faces (b) Suppose the two die were rolled many times. Approximately, what would would be the average of X? (c) Calculate the standard deviation of X.

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided die...

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided die is rolled; let Y equal the outcome (1, 2, 3, 4, 5 or 6) when a fair six-sided die is rolled. Let W=X+Y. a. What is the pdf of W? b What is E(W)?

Assume we roll a fair four-sided die marked with 1, 2, 3 and 4. (a) Find...

Assume we roll a fair four-sided die marked with 1, 2, 3 and 4. (a) Find the probability that the outcome 1 is first observed after 5 rolls. (b) Find the expected number of rolls until outcomes 1 and 2 are both observed. (c) Find the expected number of rolls until the outcome 3 is observed three times. (d) Find the probability that the outcome 3 is observed exactly three times in 10 rolls given that it is first observed...

A three-sided fair die with faces numbered 1, 2 and 3 is rolled twice. List the...

A three-sided fair die with faces numbered 1, 2 and 3 is rolled twice. List the sample space. S = b.{ List the following events and their probabilities. Write probabilities in non-reduced fractional form A = rolling doubles = { P(A)= / B = rolling a sum of 4 = { P(B)= / C = rolling a sum of 5 = { P(C)= C. Are the events A and B mutually exclusive? If yes, why? If not, why not? D.Are...

A fair four sided die has two faes numbered 0 and two faces numbered 2. Another...

A fair four sided die has two faes numbered 0 and two faces numbered 2. Another fair four sided die has its faces numbered 0,1,4, and 5. The two dice are rolled. Let X and Y be the respective outcomes of the roll. Let W = X + Y. (a) Determine the pmf of W. (b) Draw a probability histogram of the pmf of W

2 fair four-sided dice with faces numbered 1, 2, 3, 4 are rolled. Random variable D...

2 fair four-sided dice with faces numbered 1, 2, 3, 4 are rolled. Random variable D is the nonnegative difference |X2 −X1| of the numbers rolled on the 2 dice and Y is a Bernoulli random variable such that Y = 0 if the sum X1 +X2 is even and and Y = 1 if the sum is odd. a. Find joint probability mass function of D and Y. b. Are D and Y dependent or independent? How do you...

Consider rolling both a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6...

Consider rolling both a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6 together. After rolling both dice, let X denote the number appearing on the foursided die and Y the number appearing on the six-sided die. Define W = X +Y . Assume X and Y are independent. (a) Find the moment generating function for W. (b) Use the moment generating function technique to find the expectation. (c) Use the moment generating function technique to find...

Consider a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6, where X...

Consider a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6, where X is the number appearing on the four-sided die and Y is the number appearing on the six-sided die. Define W=X+Y when they are rolled together. Assuming X and Y are independent, (a) find the moment generating function for W, (b) the expectation E(W), (c) and the variance Var(W). Use the moment generating function technique to find the expectation and variance.

Assume that you have a fair 6 sided die with values {1, 2, 3, 4, 5,...

Assume that you have a fair 6 sided die with values {1, 2, 3, 4, 5, 6} and a fair 12 sided die with values {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. A discrete random variable is generated by rolling the two dice and adding the numerical results together. (a) Create a probability mass function that captures the probability of all possible values of this random variable. You may use R or draw the pmf...