Chapter 4 Notes.docx

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Western University
Statistical Sciences
Statistical Sciences 2244A/B
Jennifer Waugh

4­2 Random Variables ● Random variable: a variable that has a single numerical value, determined by chance for each outcome of a procedure ○ Discrete random variable: either has finite number of values or countable number of values where “countable” means there might be infinitely many values but can be associated with counting process ○ Continuous random variable: infinitely many values and they can be associated with measurements of a continuous scale without gaps or interruptions ● Probability distribution: graph, table, or formula that gives probability for each value of the random variable Graphs ● Probability Histogram: Is similar to the frequency histograms, but instead the y-axis shows probabilities instead of relative frequencies based on actual sample results ● The areas of the rectangles on a probability histogram are the same as the probabilities ● Requirements for a Probability Distribution: ○ Sum of P(x) = 1 (sum of all probabilities is equal to 1) ○ 0 <= P(x) <= 1 (each probability value must be between 0 and 1 inclusive) Identifying Unusual Results with Probabilities ● Remember, when it comes to calculating probabilities for events, we must try to determine whether or not a given assumption is correct or not depending on if the probability is very small or not (rare event rule) ● We need to be able to determine if events occurred due to chance or because a certain technique is effective in providing certain results ● Agood example to keep in mind is flipping a coin 1000 times and getting 501 heads ● Although this seems like a normal result, as the probability of getting heads or tails every time you flip is 0.5, thus getting at least 501 heads has a reasonable probability, the probability of getting exactly 501 heads is much smaller (0.0252) ● Despite the small probability, we do not consider 501 heads out of 1000 unusual, because getting at least 501 heads results in a probability of 0.487 which is high and reasonable in this case ● Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) is very small (0.05 or less) ● Usually low: x successes among n trials is unusually low number of successes if P(x or fewer) is very small (0.05 or less) Expected Value ● The mean of a discrete random variable is the theoretical mean outcome for infinitely many trials ● We can think of that mean as the expected value in the sense that it is the average value that we would expect to get if the trials could continue indefinitely ● Expected value: of a discrete random variable is denoted by E, and represents average value of outcomes, obtained by finding value of sum of x * P(x) ○ E is therefore equal to the population mean of a discrete random variable
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