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- https://stats.libretexts.org/Courses/Fresno_City_College/Book%3A_Business_Statistics_Customized_(OpenStax)/04%3A_Discrete_Random_Variables/4.01%3A_IntroductionTo use the combinatorial formula we would solve the formula as follows: \[\left(\begin{array}{l}{4} \\ {2}\end{array}\right)=\frac{4 !}{(4-2) ! 2 !}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 ...To use the combinatorial formula we would solve the formula as follows: \[\left(\begin{array}{l}{4} \\ {2}\end{array}\right)=\frac{4 !}{(4-2) ! 2 !}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 \cdot 1}=6\nonumber\] If we wanted to know the number of unique 5 card poker hands that could be created from a 52 card deck we simply compute: \[\left(\begin{array}{c}{52} \\ {5}\end{array}\right)\nonumber\] where 52 is the total number of unique elements from which we are drawing and 5 is the siz…
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/BFE_1201_Statistical_Methods_for_Finance_(Kuter)/04%3A_Random_Variables/4.05%3A_Introduction_to_Continuous_Random_Variables/4.5.01%3A_Properties_of_Continuous_Probability_Density_FunctionsMathematically, the cumulative probability density function is the integral of the pdf, and the probability between two values of a continuous random variable will be the integral of the pdf between t...Mathematically, the cumulative probability density function is the integral of the pdf, and the probability between two values of a continuous random variable will be the integral of the pdf between these two values: the area under the curve between these values. \(P(c < X < d)\) is the probability that the random variable X is in the interval between the values c and d. \(P(c < X < d)\) is the area under the curve, above the x-axis, to the right of \(c\) and the left of \(d\).
- https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/05%3A_Continuous_Random_Variables/5.01%3A_Properties_of_Continuous_Probability_Density_FunctionsThis page discusses continuous probability distributions, highlighting the probability density function (pdf) and the cumulative distribution function (cdf) for evaluating probabilities as areas. It n...This page discusses continuous probability distributions, highlighting the probability density function (pdf) and the cumulative distribution function (cdf) for evaluating probabilities as areas. It notes that probabilities for specific values are zero and emphasizes intervals, with the total pdf area equaling one. The text covers various continuous distributions, including uniform, exponential, and normal, and offers examples for calculating probabilities within specified ranges.
- https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/04%3A_Discrete_Random_Variables/4.00%3A_Introduction_to_Discrete_Random_VariablesThis page explains probability and discrete random variables using examples like quiz performances and phone call counts. It defines random variables, which take whole number values for experimental o...This page explains probability and discrete random variables using examples like quiz performances and phone call counts. It defines random variables, which take whole number values for experimental outcomes, and introduces probability density functions (PDFs) along with combinatorial formulas for efficient probability calculation. Key mathematical concepts, including the binomial coefficient and combinatorial formulas, are emphasized in the context of these calculations.
- https://stats.libretexts.org/Courses/Fresno_City_College/Book%3A_Business_Statistics_Customized_(OpenStax)/05%3A_Continuous_Random_Variables/5.02%3A_Properties_of_Continuous_Probability_Density_FunctionsMathematically, the cumulative probability density function is the integral of the pdf, and the probability between two values of a continuous random variable will be the integral of the pdf between t...Mathematically, the cumulative probability density function is the integral of the pdf, and the probability between two values of a continuous random variable will be the integral of the pdf between these two values: the area under the curve between these values. \(P(c < x < d)\) is the probability that the random variable X is in the interval between the values c and d. \(P(c < x < d)\) is the area under the curve, above the x-axis, to the right of \(c\) and the left of \(d\).
- https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/BFE_1201_Statistical_Methods_for_Finance_(Kuter)/04%3A_Random_Variables/4.01%3A_Introduction_to_Discrete_Random_VariablesAn alternative to listing the complete sample space and counting the number of elements we are interested in, is to skip the step of listing the sample space, and simply figuring out the number of ele...An alternative to listing the complete sample space and counting the number of elements we are interested in, is to skip the step of listing the sample space, and simply figuring out the number of elements in it and doing the appropriate division.
- https://stats.libretexts.org/Courses/Fresno_City_College/Introduction_to_Business_Statistics_-_OER_-_Spring_2023/05%3A_Continuous_Random_Variables/5.02%3A_Properties_of_Continuous_Probability_Density_FunctionsMathematically, the cumulative probability density function is the integral of the pdf, and the probability between two values of a continuous random variable will be the integral of the pdf between t...Mathematically, the cumulative probability density function is the integral of the pdf, and the probability between two values of a continuous random variable will be the integral of the pdf between these two values: the area under the curve between these values. \(P(c < x < d)\) is the probability that the random variable X is in the interval between the values c and d. \(P(c < x < d)\) is the area under the curve, above the x-axis, to the right of \(c\) and the left of \(d\).
- https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/04%3A_Discrete_Random_Variables/4.01%3A_Hypergeometric_DistributionThis page discusses the hypergeometric probability distribution, used when probabilities change with each draw, such as drawing cards from a deck without replacement. It details the formula for calcul...This page discusses the hypergeometric probability distribution, used when probabilities change with each draw, such as drawing cards from a deck without replacement. It details the formula for calculating probabilities of specific outcomes, exemplified by drawing two aces in a poker hand. The page also outlines the conditions for applying the hypergeometric distribution and includes an example of calculating the probability of selecting gumdrops from a mix of candies.
- https://stats.libretexts.org/Courses/Fresno_City_College/Introduction_to_Business_Statistics_-_OER_-_Spring_2023/04%3A_Discrete_Random_Variables/4.01%3A_Introduction_to_Discrete_Random_VariablesTo use the combinatorial formula we would solve the formula as follows: \[\left(\begin{array}{l}{4} \\ {2}\end{array}\right)=\frac{4 !}{(4-2) ! 2 !}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 ...To use the combinatorial formula we would solve the formula as follows: \[\left(\begin{array}{l}{4} \\ {2}\end{array}\right)=\frac{4 !}{(4-2) ! 2 !}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 \cdot 1}=6\nonumber\] If we wanted to know the number of unique 5 card poker hands that could be created from a 52 card deck we simply compute: \[\left(\begin{array}{c}{52} \\ {5}\end{array}\right)\nonumber\] where 52 is the total number of unique elements from which we are drawing and 5 is the siz…
