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intro stat

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  1. Probability Distribution

  2. A probability distribution shows the different values a random variable can assume along with the probability that are associated with each of those value. • A random variable is a variable whose values are determined by chance. That is, it assumes a particular value associated with an outcome of an experiment. • Example: Discrete Random Variable • if a die is rolled, a letter such as X can be used to represent the outcomes. Then the value that X can assume is 1, 2, 3, 4, 5, or 6, corresponding to the outcomes of rolling a single die. • Continuous Random Variable • Variables that can assume all values in the interval between any two given values are called continuous variables. For example, if the temperature goes from 62 to 78 in a 24-hour period, it has passed through every possible number from 62 to 78. Continuous random variables are obtained from data that can be measured rather than counted.

  3. Types of Probability Distribution • A) Discrete Binomial Distribution • B) Continuous Normal Distribution • First: revisit random variable

  4. Illustration Toss 3 coins: Sample space: (outcome/sample point) Random V: (value assigned to each outcome)

  5. Illustration

  6. The following observations are true in all case about the relationship between the random variable and sample point: • Each sample point is assigned a specific possible value of the (R.V), though the same specific value can be assigned to two or more sample points. • Each possible value of (R.V) is an event since it is a subset define on a sample space. Eg: (4 has a probability of 3/36). • All the values of (R.V) constitute a set of events that are mutually exclusive and completely exhaustive.

  7. Note: • In earlier discussions, there were specific formulas used to estimate the mean, variance, and standard deviation. However, those formulas are not applicable to wider ranger of circumstances. • Our new environment will make modifications to the concepts by introducing a different approach to old problems. • In previous class, we spoke extensively of frequency distribution and various ways to address the mean, variance, STD of grouped and ungrouped data. Here, we will talk about those same measures but make slight adjustments to accommodate the random variable.

  8. Discrete Binomial Distribution • Mean, Variance, Standard Deviation, Expectation • The probability distribution of a random variable is a graph table or formula that specifies the probability associated with each possible value the random variable can assume. Mean: µ = E(x); E(x) * P(x); 12/8; = 1.5

  9. Illustration 2

  10. Binomial Distribution • Many types of probability problems have only two outcomes or can be reduced to two outcomes. • For example, when a coin is tossed, it can land heads or tails. When a baby is born, it will be either male or female. In a basketball game, a team either wins or loses. A true/false item can be answered in only two ways, true or false. • Again, a multiple-choice question, even though there are four or five answer choices, can be classified as correct or incorrect. Situations like these are called binomial experiments. Why?

  11. A binomial experiment is a probability experiment that satisfies the following four requirements: • 1. There must be a fixed number of trials. • 2. Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure. • 3. The outcomes of each trial must be independent of one another. • 4. The probability of a success must remain the same for each trial.

  12. Tossing Coins • A coin is tossed 3 times. Find the probability of getting exactly two heads. • Solution • This problem can be solved by looking at the sample space. There are three ways to get two heads. • HHH, HHT, HTH, THH, TTH, THT, HTT, TTT • The answer is 3/8 , or 0.375. • From the standpoint of a binomial experiment, one can show that this problem meets the four requirements.

  13. Survey on Employment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. https://www.youtube.com/watch?v=0NAASclUm4k

  14. Tossing Coins • A coin is tossed 3 times. Find the probability of getting exactly two heads. • Since n = 3, X = 2, and p = 0.5, the value 0.375 is found

  15. Illustration: • Survey on Fear of Being Home Alone at Night Public Opinion reported that 5% of Americans are afraid of being alone in a house at night. If a random sample of 20 Americans is selected, find these probabilities by using the binomial table. • a. There are exactly 5 people in the sample who are afraid of being alone at night. • b. There are at most 3 people in the sample who are afraid of being alone at night. • c. There are at least 3 people in the sample who are afraid of being alone at night.

  16. Tossing a Coin: A coin is tossed 4 times. Find the mean, variance, and standard deviation of the number of heads that will be obtained.

  17. Rolling a Die • A die is rolled 480 times. Find the mean, variance, and standard deviation of the number of 3s that will be rolled.

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