Random Variables and Probability
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Questions and Answers

Which of the following statements is NOT a characteristic of probabilities associated with a random variable?

  • Each probability value is between 0 and 1, inclusive.
  • Probabilities can be negative if the random variable represents a loss. (correct)
  • The sum of all probabilities for all possible values of the random variable equals 1.
  • Probabilities are used to describe the likelihood of each possible outcome occurring.

    • A discrete random variable can take on any value within a given range, including fractional and decimal values.

    False (B)

    What is the sum of all probabilities for all possible values of a discrete random variable?

    1

    For any random variable, $0 ≤ P(x) ≤ ______$, indicating that probabilities cannot exceed this value.

    <p>1</p> Signup and view all the answers

    Match the random variables with their corresponding value types.

    <p>Number of heads in multiple coin flips = Discrete Time it takes sprinters to run 100 meters = Continuous Weight of baby elephants = Continuous Number of college graduates in the households = Discrete</p> Signup and view all the answers

    Which of the following real-world examples would be best represented by a continuous random variable, rather than a discrete one?

    <p>The height of a randomly selected basketball player. (B)</p> Signup and view all the answers

    Define the expected value of a discrete random variable.

    <p>The mean of a discrete random variable X.</p> Signup and view all the answers

    The variance of a discrete random variable can be negative.

    <p>False (B)</p> Signup and view all the answers

    Which of the following best describes the focus of Political Science according to the provided content?

    <p>The systematic study of politics and how people make, preserve, and amend the rules under which they live. (C)</p> Signup and view all the answers

    Traditions are defined solely by material objects and possessions accumulated over time.

    <p>False (B)</p> Signup and view all the answers

    How does culture influence a society, according to the content provided?

    <p>Culture is the big picture; it's the overall way of life of a group of people. It influences a society through beliefs, ideas, values, practices, knowledge, history, shared experiences, and attitudes.</p> Signup and view all the answers

    According to the content, a ______ is an object, word, or action that stands for something else, without natural relationship, that is culturally defined.

    <p>symbol</p> Signup and view all the answers

    Match each characteristic with its correct descriptor related to the concept of culture:

    <p>Culture is social = It is a product of learned behavior. Culture varies = It differs from one society to another. Culture is shared = It is commonly held among members of a society. Culture is learned = It is acquired and transmitted across generations.</p> Signup and view all the answers

    Which anthropological focus studies human origins, evolution, adaptation, and variation?

    <p>Biological Anthropology (B)</p> Signup and view all the answers

    Which concept is described as the activity through which people make, preserve, and amend the general rules under which they live?

    <p>Politics (C)</p> Signup and view all the answers

    Customs are broad, abstract principles that guide thoughts and actions rather than specific practices.

    <p>False (B)</p> Signup and view all the answers

    Which of the following statements accurately describes a key property of the total area under a normal distribution curve?

    <p>The total area under the curve is 1, representing 100% of the probability. (D)</p> Signup and view all the answers

    In a standard normal distribution, the mean ($\mu$) is equal to 1 and the standard deviation ($\sigma$) is equal to 0.

    <p>False (B)</p> Signup and view all the answers

    What is the purpose of calculating a z-score?

    <p>To determine the number of standard deviations a data point is from the mean.</p> Signup and view all the answers

    A curve is said to be ______ to the x-axis if it approaches the x-axis but never actually touches or intersects it.

    <p>asymptotic</p> Signup and view all the answers

    In the z-score formula for a sample, what does 's' represent?

    <p>Standard deviation (A)</p> Signup and view all the answers

    Which of the following is the correct formula for calculating the z-score for a population?

    <p>$z = \frac{x - \mu}{\sigma}$ (B)</p> Signup and view all the answers

    A student scores 80 on a test where the mean is 70 and the standard deviation is 5. What is the student’s z-score?

    <p>2 (D)</p> Signup and view all the answers

    Match the following terms with their corresponding definitions:

    <p>Z-score = Indicates how many standard deviations an element is from the mean Standard Normal Distribution = A normal distribution with a mean of 0 and a standard deviation of 1 Population Mean ($\mu$) = The average value of a population Standard Deviation ($\sigma$) = A measure of the spread of data around the mean</p> Signup and view all the answers

    Which condition most directly contradicts the Hardy-Weinberg principle's assumption of random mating and equal embryo survival probability?

    <p>Habitat fragmentation affecting mate access. (C)</p> Signup and view all the answers

    If a population splits due to vicariance, the newly isolated sub-populations are likely to maintain identical allele frequencies to the original population.

    <p>False (B)</p> Signup and view all the answers

    How might rare genes in a founding population impact the adaptation of organisms to a new isolated environment?

    <p>Rare genes can become more prevalent, leading to distinct behavioral or morphological variations that enhance survival in the new environment.</p> Signup and view all the answers

    The evolutionary process where new species arise from a single ancestral species within the same geographic area is known as ______ speciation.

    <p>sympatric</p> Signup and view all the answers

    Match the following evolutionary mechanisms with their primary effect on population genetics:

    <p>Habitat Fragmentation = Reduction in gene flow, leading to potential genetic divergence Vicariance = Separation of populations by a physical barrier Founder Effect = Change in allele frequencies due to a small number of individuals establishing a new population Sympatric Speciation = Formation of new species within the same geographic area</p> Signup and view all the answers

    Which of the following characteristics accurately describe plasmids used in gene cloning?

    <p>Small, circular DNA with a single origin of replication (B)</p> Signup and view all the answers

    Polymerase Chain Reaction (PCR) is a manual method requiring constant researcher intervention to amplify specific DNA sequences.

    <p>False (B)</p> Signup and view all the answers

    What is the primary role of restriction enzymes in the process of gene isolation and cloning?

    <p>Restriction enzymes cut DNA at specific sites.</p> Signup and view all the answers

    During PCR, the solution is ______ to break the hydrogen bonds between base pairs, causing the DNA double helix to open.

    <p>heated</p> Signup and view all the answers

    What is the role of DNA ligase in the context of recombinant DNA (rDNA) technology?

    <p>To 'paste' or join DNA fragments together (A)</p> Signup and view all the answers

    Match the following steps with their corresponding function in the PCR process:

    <p>Heating the solution = Breaks the hydrogen bonds. Cooling the solution = Allows the primers to attach to the DNA sequence.</p> Signup and view all the answers

    In the context of gene cloning, what is a vector?

    <p>A carrier molecule that transfers rDNA into a host cell (A)</p> Signup and view all the answers

    Briefly outline the three main steps involved in a standard Polymerase Chain Reaction (PCR) cycle.

    <p>Denaturation, annealing, and extension.</p> Signup and view all the answers

    Which of the following scenarios would most likely disrupt Hardy-Weinberg equilibrium in a population?

    <p>Consistent emigration of individuals with a specific trait from the population. (A)</p> Signup and view all the answers

    According to Lamarck's theory, acquired characteristics are inherited because genetic material changes in response to environmental pressures.

    <p>False (B)</p> Signup and view all the answers

    In the Hardy-Weinberg equation, what do the terms $p^2$, $2pq$, and $q^2$ represent?

    <p>$p^2$ represents the frequency of homozygous dominant individuals, $2pq$ represents the frequency of heterozygous individuals, and $q^2$ represents the frequency of homozygous recessive individuals.</p> Signup and view all the answers

    Darwin's observations of distinct flora and fauna across different regions led him to realize that life forms ______ over time and vary from place to place.

    <p>change</p> Signup and view all the answers

    Match the following concepts with their descriptions:

    <p>Hardy-Weinberg Equilibrium = A principle stating that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. Lamarck's Theory = An evolutionary idea suggesting organisms could pass on traits acquired during their lifetime to their offspring. Darwin's Observation = The recognition that flora and fauna differ in different regions.</p> Signup and view all the answers

    If a population of butterflies has two alleles for wing color, $B$ (dominant, black) and $b$ (recessive, white), and the frequency of the $b$ allele is 0.4, what is the frequency of the heterozygous genotype ($Bb$) assuming the population is in Hardy-Weinberg equilibrium?

    <p>0.48 (D)</p> Signup and view all the answers

    Lamarck's theory of inheritance of acquired characteristics was ultimately rejected because:

    <p>There was no evidence that changes in somatic cells (body cells) directly alter the genetic material passed on to offspring. (B)</p> Signup and view all the answers

    In a population, the observed frequency of the homozygous recessive genotype is 0.04. Assuming Hardy-Weinberg equilibrium, what is the predicted frequency of the dominant allele?

    <p>0.8 (D)</p> Signup and view all the answers

    Flashcards

    Probability Range

    The probability of each value is between 0 and 1.

    Random Variable

    A variable whose values depend on probability outcomes.

    Probability Sum

    The sum of all probabilities in a distribution equals 1.

    Mean (Expected Value)

    The average or expected value of a random variable.

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    Variance

    A measure of how much the values of a random variable differ from the mean.

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    Discrete Random Variable

    A random variable that can take on a countable number of values.

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    Examples of Random Variables

    Events like the weight of elephants or sports times give specific values.

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    Probability Notation

    P(x) denotes the probability of a specific outcome x.

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    Standard Normal Distribution

    A normal distribution with mean (μ) = 0 and standard deviation (σ) = 1.

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    Z-Score

    A statistic that tells how many standard deviations a data point is from the mean.

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    Formula for Z-Score (Population)

    z = (x - μ) / σ; where x is raw score, μ is population mean, σ is population standard deviation.

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    Formula for Z-Score (Sample)

    z = (x - x̄) / s; where x̄ is sample mean and s is sample standard deviation.

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    Asymptotic Curve

    A curve that approaches a line but never touches it, typically the x-axis in normal distribution.

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    Total Area under Curve

    In normal distributions, the total area equals 1 or 100%, representing total probability.

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    Finding P-value

    The probability associated with a specific z-score found using the z-table.

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    Using Z-table

    A table that provides the probability of a given z-score in a standard normal distribution.

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    Culture

    The overall way of life of a group, including beliefs, values, and practices.

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    Beliefs

    Specific ideas that people hold to be true within a culture.

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    Traditions

    Customs or practices passed down from one generation to another.

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    Customs

    Specific ways of doing things observed by a group.

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    Symbol

    An object, word, or action that stands for something else, defined by culture.

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    Variation in Culture

    Culture differs from society to society based on beliefs and practices.

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    Learned Culture

    Culture is taught and learned, not inherited biologically.

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    Cultural Anthropology

    The study of origins, beliefs, and practices of humans within societies.

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    Hardy-Weinberg principle

    A principle stating that allele frequencies in a population remain constant in the absence of evolutionary influences.

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    Gamete frequencies

    The ratio of different alleles in gametes produced by a population.

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    Sympatric speciation

    The evolution of new species from a common ancestor while living in the same geographic area.

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    Population allele frequency

    The proportion of a specific allele among all allele copies in a population.

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    Plasmid

    A small, circular DNA molecule that usually carries one or a few genes.

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    Restriction Enzymes

    Proteins that cut DNA at specific sequences to isolate gene fragments.

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    DNA Ligase

    An enzyme that joins DNA fragments together after cutting.

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    Polymerase Chain Reaction (PCR)

    A technique to amplify copies of specific DNA sequences quickly.

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    Vectors

    Molecules (often plasmids) used to transfer DNA into host cells.

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    Gene Isolation

    The process of extracting a specific gene from a DNA library.

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    Electrophoresis

    A technique that separates DNA fragments based on size using an electric field.

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    Thermal Cycling

    The process of heating and cooling in PCR to amplify DNA.

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    Hardy-Weinberg Equilibrium

    A model predicting genetic stability in a population across generations.

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    Hardy-Weinberg Equation

    p² + 2pq + q² = 1, describes allele frequencies in a population.

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    Allele Frequency

    The proportion of a specific allele among all allele copies in a population.

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    Jean-Baptiste Lamarck

    Naturalist who proposed that organisms evolve through acquired characteristics.

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    Acquired Characteristics

    Traits gained during an organism's lifetime that can supposedly be inherited.

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    Charles Darwin

    Naturalist known for his theory of evolution and natural selection.

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    Evidence of Evolution

    Darwin's observations showed distinct species in different regions, supporting adaptation.

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    Field Mice Example

    Application of Hardy-Weinberg to predict allele frequencies in a real population.

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    Study Notes

    Statistics and Probability

    • A random variable is a variable whose possible values are determined by chance
    • It's usually represented by an uppercase letter (e.g., X)
    • A lowercase letter (e.g., x) represents one of its values
    • A discrete random variable can only take a finite (countable) number of distinct values
    • These values are exact and can be represented by nonnegative whole numbers
    • Examples of discrete variables include the number of books in a bookshelf, the number of players in a race, and the number of students in a class
    • A continuous random variable can take on an infinite number of values in an interval between two specific values
    • Examples of continuous variables include time, size, weight, and temperature

    Probability Distribution of Discrete Variables

    • A listing of all possible values of a discrete random variable along with their corresponding probabilities is called a discrete probability distribution
    • The probabilities for each value must be between 0 and 1
    • The sum of all probabilities must equal 1
    • Probability distributions can be presented in tabular, graphical, or formula form

    Mean and Variance of Discrete Variables

    • The mean of a discrete random variable X is also called its expected value
    • It's the weighted average of all the values that the random variable X would assume in the long run
    • The formula for the expected value is E(X) = Σ [x * P(x)]
    • The variance of a discrete random variable X is denoted by σ²
    • The formula for the variance is σ² = Σ[(x − μ)² * P(x)], where μ is the population mean
    • The standard deviation is the square root of the variance, σ

    Normal Distribution

    • A continuous random variable that has a bell-shaped probability distribution
    • The mean, median, and mode are equal and located at the center of the distribution
    • Z-scores help to indicate the area/probability under the normal curve.

    Sampling Techniques

    • Simple Random Sampling: Every element in the population has an equal chance of being selected for the sample.
    • Systematic Random Sampling: Every nth element of the population is selected. The value of n is calculated by dividing the number of elements in the population by the number of elements in the desired sample.
    • Stratified Random Sampling: The population is divided into strata, and then samples are randomly selected separately from each stratum.
    • Cluster Sampling: The entire population is broken into small groups (clusters), and then some of the clusters are randomly selected.

    Practical Research

    • Research involves asking questions and seeking answers to achieve a greater understanding of a certain topic
    • Inquiry involves actively seeking, truth, information, or knowledge
    • Research is essential for addressing problems and promoting improvements in the standard of living in developing countries.

    Genetic Engineering

    • Genetic engineering is used to manipulate genes in organisms to create beneficial products.
    • It leads to increased crop productivity through introducing disease-resistance traits or drought tolerance.
    • An example of this includes producing papaya resistant to the Papaya ringspot virus.
    • Genetic engineering also has potential applications in microbial bioremediation, using microorganisms to break down harmful environmental contaminants

    Biotechnology

    • Biotechnology involves manipulating biological processes to produce useful products
    • Recombinant DNA is artificially created DNA that combines DNA from different sources and is used in cloning processes
    • Restriction enzymes are used to cut DNA in specific places
    • Plasmids are used as vectors to transfer and replicate recombinant DNA in host cells

    Evidences of Evolution

    • The fossil record shows a progression of life forms over time.
    • The geologic time scale displays the sequence of life forms over time.
    • Comparing anatomy shows structural similarities indicating a shared evolutionary origin.
    • Comparing geographical distribution of species reveals the adaptation of species to their environment.
    • The study of heredity (genetics) helps understand populations and the way they change over time.

    Speciation

    • Speciation is the process by which new species arise.
    • Allopatric speciation is when populations are geographically isolated
    • Peripatric speciation is a form of allopatric speciation involving isolation of a small subpopulation
    • Sympatric speciation happens when species arise in the same geographic area
    • Parapatric speciation occurs in different environments within a population where the conditions are slightly different

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