(a) construct a discrete probability distribution for the...

(a) construct a discrete probability distribution for the random variable $X$ /Hint: $P\left(x_{i}\right)=\frac{f_{i}}{N} .$, (b) draw the probability histogram, (c) compute and interpret the mean of the random variable $X$, and (d) compute the standard deviation of the random variable $X$. The following data represent the number of games played in each World Series from 1923 to 2005.

(a) construct a discrete probability distribution for the random variable $X$ /Hint: $P\left(x_{i}\right)=\frac{f_{i}}{N} .$, (b) draw the probability histogram, (c) compute and interpret the mean of the random variable $X$, and (d) compute the standard deviation of the random variable $X$.
The following data represent the number of games played in each World Series from 1923 to 2005.

Key Concepts

Mean of a Random Variable

The mean, or expected value, of a random variable is a measure of central tendency that represents the long-run average outcome of a random process. It is calculated as the weighted sum of all possible values, with each value weighted by its respective probability. The mean provides insight into the typical outcome one can expect from the probability distribution.

Probability Histogram

A probability histogram is a graphical representation of a discrete probability distribution. Each bar in the histogram corresponds to a possible outcome of the random variable, and the height of the bar represents the probability of that outcome. This visual tool helps in understanding the distribution of the data by showing the spread and frequency of each outcome.

Standard Deviation of a Random Variable

The standard deviation measures the dispersion or spread of a random variable's values around the mean. It is the square root of the variance, which is the average of the squared differences between each outcome and the mean. This concept is crucial for assessing the variability in the data and understanding the reliability of the mean as a summary statistic.

Discrete Probability Distribution

A discrete probability distribution assigns probabilities to each possible value of a discrete random variable. It is often constructed using the relative frequency of occurrence for each outcome, where each probability is computed as the frequency of that outcome divided by the total number of observations. This concept is fundamental in statistics for modeling and analyzing data where outcomes are countable and distinct.

Relative Frequency

Relative frequency is a method of determining the probability of an event by dividing the number of times the event occurs by the total number of observations or trials. This approach is commonly used to estimate probabilities when theoretical models are not available and is essential for constructing empirical probability distributions from data.

Transcript

00:01 The following is a solution to number 23, and this looks at the number of games played in each world series from 1923 to 2005.

00:07 So four games that happened, so the world series got done in four games that happened for 16 world series.

00:15 Five games was 15 world series, six games was 18 world series, and then obviously they usually win seven games.

00:20 That's 33 world series.

00:22 And we're asked to find the probability distribution or created probability distribution.

00:27 So we need the total number first.

00:29 So if we were to add these together, these frequencies 16 plus 15 plus 18 plus 33, we get 82 years worth of world series.

00:37 So then we just take the frequency divided by the 82 to get the probability.

00:43 And whenever we do that, we go line here, we get 0 .195.

00:49 So that's 16 divided by 82 is 0 .195.

00:52 15 divided by 82 is 0 .183 and then 0 .22 and then 0 .40.

00:59 Two.

01:00 Another good way to double check is, so this is your probability distribution.

01:05 These should add up to pretty close to one.

01:08 Now, there may be a little off due rounding, but mine was one.

01:12 Okay, and then part b is to make a histogram with those probabilities.

01:16 So i kind of already set it up here.

01:18 So the four was about point two...

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