STATISTICS 125 - CHAPTER 6.1 Discrete Random Variables

8 September 2022
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question
What are the two requirements for a discrete probability distribution?
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The first rule states that the sum of the probabilities must equal 1. The second rule states that each probability must be between 0 and 1, inclusive. Let P(x) = 1 1. ‚ąĎ P(x) = 1 2. 0 ‚ȧ P(x) ‚ȧ
question
Determine whether the random variable is discrete or continuous. In each‚Äč case, state the possible values of the random variable. ‚Äč(a) The number of points scored during a basketball game. ‚Äč(b) The time it takes to fly from City Upper A to City Upper B.
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(a) The random variable is discrete. The possible values are x = 0, ‚Äč1, ‚Äč2,.... (b) The random variable is continuous. The possible values are t > 0.
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Determine whether the random variable is discrete or continuous. In each‚Äč case, state the possible values of the random variable. ‚Äč(a) The number of people in a restaurant that has a capacity of 300. ‚Äč(b) The distance a baseball travels in the air after being hit.
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(a) The random variable is discrete. The possible values are x=‚Äč0, ‚Äč1, ‚Äč2,...comma 300. (b)The random variable is continuous. The possible values are d > 0.
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Is the distribution a discrete probability‚Äč distribution? x P(x) 0 0.07 1 0.34 2 0.27 3 0.15 4 0.17
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Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and‚Äč 1, inclusive.
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Determine the required value of the missing probability to make the distribution a discrete probability distribution. x P(x) 3 0.34 4 ? 5 0.08 6 0.29
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In a discrete probability‚Äč distribution, the sum of the probabilities must equal‚Äč 1, and all probabilities must be greater than or equal to 0 and less than or equal to 1. Notice that all the given probabilities are greater than or equal to 0 and less than or equal to 1. The probability‚Äč P(4) is missing from the distribution. To find‚Äč P(4), first add all of the given probabilities. ‚ąĎ P(4) = 0.34 + 0.08 + 0.29 = 0.71 Recall that the sum of all the probabilities must equal 1 in a discrete probability distribution. To find‚Äč P(4), subtract the sum of the other probabilities from 1. 1.00 - 0.71 = 0.29 The value for‚Äč P(4) is a valid probability because it is greater than or equal to 0 and less than or equal to 1. Thus, ‚ÄčP(4)=0.29 makes the probability distribution valid.
question
In the probability distribution to the‚Äč right, the random variable X represents the number of marriages an individual aged 15 years or older has been involved in. Complete parts‚Äč (a) through‚Äč (f) below. x P(x) 0 0.263 1 0.576 2 0.127 3 0.029 4 0.004 5 0.001
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(a) Verify that this is a discrete probability distribution. In a discrete probability‚Äč distribution, all of the probabilities are between 0 and‚Äč 1, inclusive, and the sum of the probabilities is 1. Identify the smallest probability in this distribution. The smallest probability is 0.001. Identify the greatest probability in this distribution. The greatest probability is 0.576. So all of the probabilities are between 0 and‚Äč 1, inclusive. Now find the sum of the probabilities. 0.263+0.576+0.127+0.029+0.004+0.001=1 So the sum of the probabilities is 1. This verifies that this is a discrete probability distribution. (b) Draw the graph of the discrete probability distribution. Describe the shape of the distribution. The distribution has one mode and is skewed right.