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BR SIDDHARTH

BR S.

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In the Powerball lottery, contestants pick 5 different integers between 1 and 45 , and in addition, pick a bonus integer from the same range (the bonus integer can equal one of the first five integers chosen). Some contestants choose the numbers themselves, and others let the computer choose the numbers. The data shown in Table 5.7 are the contestant-chosen numbers in a certain state on May $3,1996 .$ A spike graph of the data is shown in Figure 5.5 . The goal of this problem is to check the hypothesis that the chosen numbers are uniformly distributed. To do this, compute the value $v$ of the random variable $\chi^{2}$ given in Example $5.6 .$ In the present case, this random variable has 44 degrees of freedom. One can find, in a $\chi^{2}$ table, the value $v_{0}=59.43$, which represents a number with the property that a $\chi^{2}$ -distributed random variable takes on values that exceed $v_{0}$ only $5 \%$ of the time. Does your computed value of $v$ exceed $v_{0} ?$ If so, you should reject the hypothesis that the contestants? choices are uniformly distributed.

Introduction to Probability

Distributions and Densities

Important Distributions

When John Kemeny was chair of the Mathematics Department at Dartmouth College, he received an average of ten letters each day. On a certain weekday he received no mail and wondered if it was a holiday. To decide this he computed the probability that, in ten years, he would have at least 1 day without any mail. He assumed that the number of letters he received on a given day has a Poisson distribution. What probability did he find? Hint: Apply the Poisson distribution twice. First, to find the probability that, in 3000 days, he will have at least 1 day without mail, assuming each year has about 300 days on which mail is delivered.

Introduction to Probability

Distributions and Densities

Important Distributions
Let $n$ be a positive integer. Let $S$ be the set of integers between 1 and $n$. Consider the following process: We remove a number from $S$ at random and write it down. We repeat this until $S$ is empty. The result is a permutation of the integers from 1 to $n$. Let $X$ denote this permutation. Is $X$ uniformly distributed?

Let $n$ be a positive integer. Let $S$ be the set of integers between 1 and $n$. Consider the following process: We remove a number from $S$ at random and write it down. We repeat this until $S$ is empty. The result is a permutation of the integers from 1 to $n$. Let $X$ denote this permutation. Is $X$ uniformly distributed?

Introduction to Probability

Distributions and Densities

Important Distributions
For which of the following random variables would it be appropriate to assign a uniform distribution? (a) Let $X$ represent the roll of one die. (b) Let $X$ represent the number of heads obtained in three tosses of a coin. (c) A roulette wheel has 38 possible outcomes: $0,00,$ and 1 through $36 .$ Let $X$ represent the outcome when a roulette wheel is spun. (d) Let $X$ represent the birthday of a randomly chosen person. (e) Let $X$ represent the number of tosses of a coin necessary to achieve a head for the first time.

For which of the following random variables would it be appropriate to assign a uniform distribution? (a) Let $X$ represent the roll of one die. (b) Let $X$ represent the number of heads obtained in three tosses of a coin. (c) A roulette wheel has 38 possible outcomes: $0,00,$ and 1 through $36 .$ Let $X$ represent the outcome when a roulette wheel is spun. (d) Let $X$ represent the birthday of a randomly chosen person. (e) Let $X$ represent the number of tosses of a coin necessary to achieve a head for the first time.

Introduction to Probability

Distributions and Densities

Important Distributions

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Caroline Mcclure verified

Numerade educator

An independent research organization is trying to estimate the probability that an accident at a nuclear power plant will result in radiation leakage. The types of accidents possible at the plant are, fire hazards, mechanical failure, or human error. The research organization also knows that two or more types of accidents cannot occur simultaneously. According to the studies carried out by the organization, the probability of a radiation leak in case of a fire is 20%, the probability of a radiation leak in case of a mechanical 50%, and the probability of a radiation leak in case of a human error is 10%. The studies also showed the following; The probability of a radiation leak occurring simultaneously with a fire is 0.1%. The probability of a radiation leak occurring simultaneously with a mechanical failure is 0.15%. The probability of a radiation leak occurring simultaneously with a human error is 0.12%. On the basis of the information available, answer the questions below: 2.1 What are the probabilities of a fire, a mechanical failure, and a human error respectively? 2.2 What is the probability of a radiation leak? 2.3 Suppose there has been a radiation leak in the reactor for which the definite cause is not known. What is the probability that it has been caused by:

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