Probability Density Function and CDF
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Questions and Answers

What is the value of k?

2

What is the formula for the cumulative density function F(x)?

F(x) = ∫ f(t) dt

What is the formula for F(x) for 0 ≤ x ≤ 1?

x²

What is the formula for F(x) elsewhere?

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Flashcards

Probability Density Function (PDF)

A function that describes the probability of a continuous random variable taking on a specific value.

Cumulative Density Function (CDF)

A function that describes the cumulative probability of a continuous random variable taking on a value less than or equal to a given value.

Normalization Condition for Continuous Random Variables

The integral of the probability density function from negative infinity to positive infinity must equal 1. This ensures that the total probability of all possible values of a variable is 1.

Normalization Condition for Discrete Random Variables

The sum of the probabilities of all possible outcomes of a discrete random variable must equal 1.

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Discrete Random Variable

A random variable that can take on a finite number of values or a countably infinite number of values.

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Continuous Random Variable

A random variable that can take on any value within a continuous range.

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Expected Value (E[X])

The expected value of a random variable, calculated as the weighted average of its possible values, weighted by their respective probabilities.

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Variance (Var[X])

The variance of a random variable, calculated as the expected value of the squared deviation from the mean.

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Standard Deviation (SD[X])

The standard deviation of a random variable, calculated as the square root of the variance.

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Conditional Probability (P(A|B))

The probability of an event occurring given that another event has already occurred.

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Independence of Events

Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.

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Joint Probability Distribution Function (f(x,y))

A function that describes the probability of two random variables, X and Y, taking on specific values simultaneously.

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Marginal Probability Distribution

The probability distribution of a single random variable, calculated by summing the joint probabilities over all possible values of the other variable.

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Conditional Probability (P(A|B)

The probability of an event occurring given that another event has occurred, calculated as the ratio of the joint probability of the two events to the marginal probability of the event that has already occurred.

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Discrete Random Variable

A random variable that takes on values based on the outcome of a specific experiment.

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Probability Mass Function (PMF)

A function that assigns probabilities to each possible value of a discrete random variable.

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Cumulative Distribution Function (CDF)

A function that describes the probability of a random variable taking on a value less than or equal to a given value.

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Expected Value (E[X])

The expected value of a random variable, calculated as the weighted average of its possible values, weighted by their respective probabilities.

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Variance (Var[X])

The variance of a random variable, calculated as the expected value of the squared deviation from the mean.

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Standard Deviation (SD[X])

The standard deviation of a random variable, calculated as the square root of the variance.

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Conditional Probability (P(A|B))

The probability of an event occurring given that another event has already occurred.

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Independence of Events

Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.

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Joint Probability Distribution Function (f(x,y))

A function that describes the probability of two random variables, X and Y, taking on specific values simultaneously.

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Marginal Probability Distribution

The probability distribution of a single random variable, calculated by summing the joint probabilities over all possible values of the other variable.

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Conditional Probability (P(A|B))

The probability of an event occurring given that another event has already occurred.

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Conditional Probability Formula

The ratio of the joint probability of two events to the marginal probability of the event that has already occurred.

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Expected Value of a Function of a Random Variable

The expectation of a function of a random variable, calculated by summing the product of the function value and the probability of each possible value of the variable.

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Probability Distribution

A mathematical function that describes the probability of a random variable taking on a specific value or a range of values.

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Poisson Distribution

A distribution that describes the probability of a series of independent events occurring in a fixed amount of time or space.

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Binomial Distribution

The distribution of the number of successes in a sequence of independent Bernoulli trials with the same probability of success on each trial.

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Discrete Probability Distribution

A distribution that describes the probability of a random variable taking on a specific value given a fixed number of trials and a constant probability of success on each trial.

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Classical Probability

The probability of an event occurring is equal to the number of favorable outcomes divided by the total number of possible outcomes.

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Empirical Probability

The probability of an event occurring is calculated by observing the event over a long period or a large number of trials.

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Subjective Probability

The probability of an event occurring is based on the subjective beliefs or opinions of an individual.

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Probability Tree Diagram

A method used to find the probability of a complex event by breaking it down into simpler events that are easier to calculate.

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Probability of an Event

The ratio of the number of favorable outcomes to the total number of possible outcomes.

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Probability Axiom

The sum of the probabilities of all possible outcomes of a random variable must equal 1.

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Study Notes

Finding k

  • The function f(x) is defined as follows: f(x) = kx, if 0 ≤ x ≤ 1; 0, otherwise
  • For the function to be a probability density function, the integral of f(x) over all possible values of x must equal 1.
  • Thus, the integral of f(x) from negative infinity to positive infinity must equal 1.
  • ∫f(x) dx = ∫10 kx dx = 1
  • Evaluating the integral: k[x2/2]10 = (k(1)2 / 2 ) - (k(0)2 / 2) = k/2 = 1
  • Solving for k: k = 2

Finding F(x)

  • F(x) represents the cumulative distribution function (CDF) related to the probability density function (PDF) denoted by f(x).
  • F(x) is defined through integration: F(x) = ∫x-∞ f(t) dt.
  • Since f(t) is defined in segments: F(x) = ∫x0 2t dt when 0 ≤ x ≤ 1 and 0 otherwise
  • Evaluating the integral ∫x0 2t dt = t2 |x0 = x2
  • Thus, the resulting CDF function is: F(x) = x2, for 0 ≤ x ≤ 1; 0, otherwise

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Description

This quiz explores the concepts of probability density functions (PDF) and cumulative distribution functions (CDF) through the function f(x) = kx. It includes topics such as integration of functions and finding constant values for PDFs. Test your understanding of these fundamental concepts in probability theory.

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