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- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/DSCI_500B_Essential_Probability_Theory_for_Data_Science_(Kuter)/04%3A_Continuous_Random_Variables/4.02%3A_Expected_Value_and_Variance_of_Continuous_Random_VariablesThe formula for the expected value of a continuous random variable is the continuous analog of the expected value of a discrete random variable, where instead of summing over all possible values we in...The formula for the expected value of a continuous random variable is the continuous analog of the expected value of a discrete random variable, where instead of summing over all possible values we integrate (recall Sections 3.4 & 3.5). For the variance of a continuous random variable, the definition is the same and we can still use the alternative formula given by Theorem 3.5.1, only we now integrate to calculate the value:
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/DSCI_500B_Essential_Probability_Theory_for_Data_Science_(Kuter)/03%3A_Discrete_Random_Variables
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/3%3A_Discrete_Random_Variables
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/DSCI_500B_Essential_Probability_Theory_for_Data_Science_(Kuter)/06%3A_The_Sample_Mean_and_Central_Limit_Theorem
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/DSCI_500B_Essential_Probability_Theory_for_Data_Science_(Kuter)/07%3A_The_Sample_Variance_and_Other_Distributions/7.01%3A_Chi-Squared_DistributionsGiven that there is more area under the \(t\) distribution pdf curves in these extreme regions, this implies that there is a higher probability that the value of a random variable with a \(t\) distrib...Given that there is more area under the \(t\) distribution pdf curves in these extreme regions, this implies that there is a higher probability that the value of a random variable with a \(t\) distribution would be this far away from the center compared to a standard normal random variable, meaning that there is greater spread in the values from the mean and hence a larger variance for the \(t\) distribution.
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_346_-_Statistics_(Kuter)
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/BFE_1201_Statistical_Methods_for_Finance_(Kuter)/03%3A_Probability_Topics/3.07%3A_Independent_EventsConsider the context of Exercise 2.2.1, where we randomly draw a card from a standard deck of 52 and \(C\) denotes the event of drawing a club, \(K\) the event of drawing a King, and \(B\) the event o...Consider the context of Exercise 2.2.1, where we randomly draw a card from a standard deck of 52 and \(C\) denotes the event of drawing a club, \(K\) the event of drawing a King, and \(B\) the event of drawing a black card. Alternately, knowing that the card drawn is black increases the probability that the card is a club, since the proportion of clubs in the entire deck is \(1/4\), but the proportion of clubs in the collection of black cards is \(1/2\).
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/BFE_1201_Statistical_Methods_for_Finance_(Kuter)/03%3A_Probability_Topics/3.06%3A_Conditional_Probability_and_Bayes'_RuleIn computing a conditional probability we assume that we know the outcome of the experiment is in event \(B\) and then, given that additional information, we calculate the probability that the outcome...In computing a conditional probability we assume that we know the outcome of the experiment is in event \(B\) and then, given that additional information, we calculate the probability that the outcome is also in event \(A\). If we let \(C\) denote the event that the card is a club and \(K\) the event that it is a King, then we are looking to compute $$P(C\ |\ K) = \frac{P(C\cap K)}{P(K)}.\label{condproba}$$ To compute these probabilities, we count the number of outcomes in the following events:
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/2%3A_Computing_Probabilities/2.3%3A_Independent_EventsConsider the context of Exercise 2.2.1, where we randomly draw a card from a standard deck of 52 and \(C\) denotes the event of drawing a club, \(K\) the event of drawing a King, and \(B\) the event o...Consider the context of Exercise 2.2.1, where we randomly draw a card from a standard deck of 52 and \(C\) denotes the event of drawing a club, \(K\) the event of drawing a King, and \(B\) the event of drawing a black card. Alternately, knowing that the card drawn is black increases the probability that the card is a club, since the proportion of clubs in the entire deck is \(1/4\), but the proportion of clubs in the collection of black cards is \(1/2\).
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/3%3A_Discrete_Random_Variables/3.6%3A_Expected_Value_of_Discrete_Random_VariablesIn other words, if we repeat the underlying random experiment several times and take the average of the values of the random variable corresponding to the outcomes, we would get the expected value, ap...In other words, if we repeat the underlying random experiment several times and take the average of the values of the random variable corresponding to the outcomes, we would get the expected value, approximately. (Note: This interpretation of expected value is similar to the relative frequency approximation for probability discussed in Section 1.2.) Again, we see that the expected value is related to an average value of the random variable.
- https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/03%3A_Probability_Topics/3.06%3A_Conditional_Probability_and_Bayes'_RuleThis page discusses conditional probability, detailing its definition and mathematical formulation, along with relevant examples like card drawing. It covers the Multiplication Law and Law of Total Pr...This page discusses conditional probability, detailing its definition and mathematical formulation, along with relevant examples like card drawing. It covers the Multiplication Law and Law of Total Probability for calculating probabilities across multiple events, and introduces Bayes' Rule for reverse conditional probabilities.
