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Probability and Statistics Assignment: Analyzing Random Variables and Distributions

An assignment focused on analyzing random variables and distributions in probability and statistics.

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    Probability and Statistics Assignment: Analyzing Random Variables and Distributions 1) In a particular suburb, 30% of the households have installed electronic security systems. a) If a household is chosen at random from this suburb, what is the probability that this household has not installed an electronic security system? a) Probability that a randomly chosen household has not installed an electronic security system: We are told that 30% of the households have installed an electronic security system. This means that 70% of the households have not installed an electronic security system. So, the probability that a randomly chosen household has not installed an electronic security system is: Answer for part a: The probability that a household has not installed an electronic security system is 0.70 or 70%. b) If two households are chosen at random from this suburb, what is the probability that neither has installed an electronic security system? b) Probability that neither of two randomly chosen households has installed an electronic security system: The events are independent (i.e., one household’s decision does not affect the other household’s decision). The probability that one household has not installed a security system is 0.70 (from part a). So, the probability that both households have not installed a security system is: Answer for part b: The probability that neither of the two households has installed an electronic security system is 0.49 or 49%. 5) In a study designed to gauge married women’s participation in the workplace today, the data provided in the file P0 4_05.xlsx were obtained from a sample of 750 randomly selected married women. Consider a woman selected at random from this sample in answering the following questions.

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    a) What is the probability that this randomly selected woman has a job outside the home? a) Probability that the randomly selected woman has a job outside the home: 1. Find the number of women who have a job outside the home. 2. Divide this by the total number of women in the sample (750 women). b) What is the probability that this randomly selected woman has at least one child? b) Probability that the randomly selected woman has at least one child: 1. Find the number of women who have at least one child. 2. Divide this by the total number of women in the sample (750 women). c) What is the probability that this randomly selected woman has a full - time job and no more than one child? Find the number of women who have a full - time job and no more than one child. • For this, you may need to look at a combination of full - time job (F) and child count (e.g., no child or one child). Divide this by the total number of women in the sample (750 women).

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    d) What is the probability that this randomly selected woman has a part - time job or at least one child, but not both? **HINT** (n=no job, P=part time, F=full time) This part is a bit trickier because it's asking for the union of the events "part - time job" or "at least one child," but excluding the cases where both conditions are met. • Find the number of women who have a part - time job or at least one child (this is the union). • Subtract the cases where the woman has both a part - time job and at least one child (this is the intersection of the two events). This can be represented as: 10) A fair coin (i.e., heads and tails are equally likely) is tossed three times. Let X be the number of heads observed in three tosses of this fair coin. a) Find the probability distribution of X.

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    b) Find the probability that two or fewer heads are observed in three tosses.

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    c) Find the probability that at least one head is observed in three tosses. e) Find the expected value of X.

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    e) Find the standard deviation of X.

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    11) Consider a random variable with the following probability distribut ion: P(X=0) = 0.1, P(X =1) = 0.2, P(X = 2) = 0.3, P(X = 3) = 0.3, and P(X = 4) = 0.1 a) Find P(X < 2)

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