Real Random Variables Quiz

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Questions and Answers

What is the mode of a discrete random variable X, denoted as MO(X)?

  • The value in the range of X that maximizes the probability mass function X(x). (correct)
  • The average of all values in the range of X.
  • The value in the range of X that minimizes the probability mass function X(x).
  • The value in the range of X that minimizes the cumulative distribution function FX(x).

What is the condition for a function f(x) to be considered a probability density function of a continuous random variable?

  • f(x) must be non-decreasing for all x and the integral of f(x) over the entire real line must equal 1.
  • f(x) must be non-increasing for all x and the integral of f(x) over the entire real line must equal 1.
  • f(x) must be differentiable for all x and the integral of f(x) over the entire real line must equal 1.
  • f(x) must be nonnegative for all x and the integral of f(x) over the entire real line must equal 1. (correct)

What is the relationship between the probability density function (PDF) f(x) and the cumulative distribution function (CDF) F(x) of a continuous random variable?

  • F(x) is the integral of f(x).
  • f(x) is the integral of F(x).
  • f(x) is the derivative of F(x). (correct)
  • F(x) is the derivative of f(x).

What is the definition of a percentile of order 𝛼 for a random variable X with cumulative distribution function FX?

<p>A value x𝛼 such that F(x𝛼) = 𝛼. (B)</p> Signup and view all the answers

What is the median of a random variable X with cumulative distribution function FX?

<p>The value x such that FX(x) = 0.5. (A)</p> Signup and view all the answers

In the context of continuous random variables, how is the mode defined?

<p>The value that maximizes the probability density function. (D)</p> Signup and view all the answers

In the provided content, what is the relationship between the probability density function f(x) and the derivative of the cumulative distribution function F(x)?

<p>f(x) is equal to the derivative of F(x) almost everywhere, except for a finite or countable set of points. (B)</p> Signup and view all the answers

What is a common way to denote the cumulative distribution function of a random variable X?

<p>FX(x) (A)</p> Signup and view all the answers

What is the formula for calculating the expectation of a random variable X raised to the power of k, denoted as E(X^k), in the discrete case?

<p>E(X^k) = ∑x^k * P(X=x) for all x in the sample space (D)</p> Signup and view all the answers

What is the alternative way to calculate the expectation of a function of a random variable, h(X), without needing to determine its probability function?

<p>Using only the probability function of X and the function h(x) (A)</p> Signup and view all the answers

Which of the following is a valid expression for calculating E(X^2) using the probability function of X in the continuous case?

<p>E(X^2) = ∫x^2 * fX(x) dx from -∞ to +∞ (D)</p> Signup and view all the answers

What is the name given to the expectation of X raised to the power k, E(X^k), which is also known as a moment of a random variable?

<p>Moment of order k (D)</p> Signup and view all the answers

What is a centered random variable?

<p>A random variable whose expectation is zero (C)</p> Signup and view all the answers

Given a random variable X, how do you create a centered random variable using X and its expectation?

<p>X - E(X) (C)</p> Signup and view all the answers

What is the formula for calculating E(aX + b) where a and b are real numbers and X is a random variable?

<p>E(aX + b) = aE(X) + b (D)</p> Signup and view all the answers

What is the key implication of the Theorem 1 presented in the text regarding calculating the expectation of a function h(X)?

<p>It allows us to calculate the expectation of h(X) even when the probability function of X is unknown. (B)</p> Signup and view all the answers

What is the value of the moment-generating function G_X(0)?

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

For k ≥ 1, how is the k-th derivative of G_X(t) expressed?

<p>G_X(k)(t) = E(X^k e^(t.X)) (B)</p> Signup and view all the answers

What does the notation X ~ B(p) signify?

<p>X follows a Bernoulli distribution (C)</p> Signup and view all the answers

What is the implication of G_X''(t) = E(X^2 e^(t.X))?

<p>It relates the second moment of X with its moment-generating function. (D)</p> Signup and view all the answers

Which of the following statements about the Bernoulli distribution is incorrect?

<p>The probability of outcome 0 is always greater than p. (B)</p> Signup and view all the answers

Which random variable does not have an expectation due to a diverging series?

<p>A variable defined by $X(\Omega) = IN*$ (B)</p> Signup and view all the answers

What kind of probability distribution is mentioned as lacking an expectation?

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

In the formula for E(h(X)), what does h(X) represent?

<p>A real-valued function defined on the random variable (B)</p> Signup and view all the answers

What is the formula for calculating the expected value of a discrete random variable?

<p>$E(X) = \sum x P_X(x)$ (C)</p> Signup and view all the answers

For continuous random variables, what is the proper way to compute expectation if it exists?

<p>$E(X) = \int x f(x)dx$ (D)</p> Signup and view all the answers

What condition must be fulfilled for E(h(X)) to be defined?

<p>The expectation must exist (B)</p> Signup and view all the answers

In Example 8, what mathematical operation leads to the divergence of the series representing the expectation?

<p>The quadratic growth of terms (C)</p> Signup and view all the answers

What notation represents the probability law of the discrete random variable in Example 8?

<p>$\rho_X(n)$ (C)</p> Signup and view all the answers

What is the expected value E(X) for a random variable X that follows a Binomial distribution with parameters n and p?

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

Which of the following correctly represents the variance V(X) for a Binomial distributed random variable X?

<p>np(1 - p) (D)</p> Signup and view all the answers

Which expression represents the moment generating function for the Binomial random variable X?

<p>(1 - p + pe^t)^n (C)</p> Signup and view all the answers

For a Poisson distributed random variable X with parameter λ, what is the probability mass function P(X=k)?

<p>e^(-λ) λ^k / k! (D)</p> Signup and view all the answers

What relationship holds between the expected value of a random variable X following a Binomial distribution and a random variable Y following a Bernoulli distribution with parameter p?

<p>E(X) = nE(Y) (B)</p> Signup and view all the answers

What does the condition 'p ∈ (0, 1)' signify in the context of the Binomial distribution?

<p>p is a probability value between 0 and 1 (B)</p> Signup and view all the answers

In the context of moment generating functions, what does G_X''(0) represent for a Binomial random variable X?

<p>The expected value of X squared (A)</p> Signup and view all the answers

If a random variable X follows a Poisson distribution with parameter λ, what is the expected value E(X)?

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

What is the form of the function h(x) defined for the random variable Y?

<p>h(x) = ax + b where a ∈ ℝ* and b ∈ ℝ (B)</p> Signup and view all the answers

How is the probability density function f_Y(y) expressed in terms of f_X?

<p>f_Y(y) = (1/a) f_X((y - b)/a) (B)</p> Signup and view all the answers

Which approximation formula is provided for E(h(X))?

<p>E(h(X)) ≈ h(μ) + h'(μ)E(X - μ) + 1/2 h''(μ)E((X - μ)^2) (C)</p> Signup and view all the answers

What term is considered negligible when deriving the approximation formulas?

<p>Higher-order terms of o(X - μ)^2 (A)</p> Signup and view all the answers

What does the formula V(h(X)) approximate to?

<p>V(h(X)) ≈ h'(μ)V(X) (B)</p> Signup and view all the answers

Why might it not be necessary to know the probability law of Y = h(X)?

<p>Because E(Y) and V(Y) can be determined from E(X) and V(X) (C)</p> Signup and view all the answers

What is the main challenge when calculating E(h(X)) and V(h(X)) directly?

<p>It involves complex integration or summation processes (C)</p> Signup and view all the answers

When using the Taylor expansion of h(x), which terms are included in the approximation for h(x)?

<p>Terms involving first and second derivatives at μ (C)</p> Signup and view all the answers

What condition is imposed on h when considering its derivatives near μ?

<p>h must be twice differentiable near μ (C)</p> Signup and view all the answers

Flashcards

Expectation of X

A measure of the central tendency of a random variable X.

Cauchy Distribution

A continuous probability distribution that does not have a defined expectation.

Discrete Random Variable

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

Continuous Random Variable

A variable that can take on an infinite number of values within a range.

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E(h(X)) Definition

The expectation of a function h applied to random variable X, calculated differently for discrete or continuous cases.

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Expectation Calculation (Discrete)

E(X) is calculated as a sum of values weighted by their probabilities: E(X) = Σ x * P(X=x).

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Expectation Calculation (Continuous)

E(X) is calculated as the integral of x multiplied by the probability density function: E(X) = ∫ x * f(x) dx.

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Cumulative Distribution Function (CDF)

F is the CDF of random variable X, defining the probability that X will take a value less than or equal to x.

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Probability Density Function (PDF)

fX is the PDF of random variable X, which describes the likelihood of X taking on a specific value.

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Mode

The mode M O (X) is the value of X that maximizes the probability mass function in discrete variables.

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Percentile

A percentile of order α is a value xα such that the probability that X is less than or equal to xα equals α.

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Exponential Distribution

A continuous probability distribution characterized by a constant hazard rate, often represented as f(x) = e^(-x) for x ≥ 0.

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Differentiability of CDF

A CDF F is differentiable except at a countable set of points, and its derivative F' is the PDF f.

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Median

The median is the 0.5-percentile, meaning half the values are below it in a distribution.

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Probability Mass Function (PMF)

The PMF is the function that gives the probability of each outcome in a discrete random variable's possible values.

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Binomial Distribution

A probability distribution for the number of successes in a fixed number of trials.

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Expected Value (E(X))

The average outcome of a random variable, calculated using probabilities.

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Variance (V(X))

A measure of the spread of a distribution, calculating the average squared deviation from the mean.

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Generating Function (GX(t))

A function that encodes the probability distribution of a random variable.

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Poisson Distribution

A probability distribution for the number of events occurring in a fixed interval of time or space.

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E(X²)

The expected value of the square of a random variable, providing insight into the variance.

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Connections Between Distributions

If X follows Binomial(n,p) and Y follows Binomial(p), then E(X) = nE(Y).

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Parameter p in Distributions

The probability of success in a single trial in Binomial and Poisson distributions.

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Expectation of a random variable

The average value of a random variable, calculated using its probability function.

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Moment of order k

The expectation of the k-th power of a random variable, indicating its distribution characteristics.

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Centered random variable

A random variable is centered if its expectation E(X) equals zero.

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Corollary of Theorem 1

For any random variable X and constants a, b, E(aX + b) = aE(X) + b.

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Probability function

The function that describes the likelihood of various outcomes of a random variable.

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Expectation without probability function

Calculating the expectation of a function h(X) using the probability of X without directly knowing h(X)'s probabilities.

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Discrete case expectation

For discrete random variables, expectation is calculated with a sum over all outcomes.

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Continuous case expectation

For continuous random variables, expectation is calculated using integrals over a probability density function.

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Density Function

Function that describes the probability distribution of a continuous random variable.

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Function of a Random Variable

Transformation from random variable X to Y using a function h(X).

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Expectation of Y (E(Y))

The mean value of the random variable after transformation by h.

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Variance of Y (V(Y))

Measure of how much the values of Y vary after transformation by h.

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Taylor Series Expansion

A way to approximate functions using derivatives at a certain point.

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Approximation Formulas

Formulas used for estimating E(h(X)) and V(h(X)) without heavy calculations.

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First Derivative (h'(μ))

The slope of the function h at the mean value μ, indicating its rate of change.

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Second Derivative (h''(μ))

Indicates the curvature of the function at μ, showing how the rate of change itself changes.

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Higher-Order Terms

Terms in a Taylor series that are neglected for simplification when approximating functions.

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Moment-generating function (MGF)

A function that captures all moments of a random variable X, defined as M_X(t) = E(e^(tX)).

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Derivatives of MGF

The k-th derivative of the moment-generating function at t=0 gives the k-th moment of the random variable X.

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Bernoulli distribution

A discrete probability distribution for a random variable that takes value 1 with probability p and 0 with probability 1-p.

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Expectation

The average or mean value of a random variable, denoted E(X), representing the long-run average outcome.

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Variance

A measure of the spread of a random variable, calculated as the expected value of the squared deviation from the mean.

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Study Notes

Real Random Variables

  • A real random variable (RV) is a function defined on a probability space (Ω, A, P) that maps outcomes in Ω to real numbers.
  • The pre-image of any interval in the real numbers must be an event in the o-algebra A.
  • The set of possible values taken by the RV is called the image of X, denoted X(Ω).

Discrete Real Random Variables (DRV)

  • A DRV is a RV whose possible values form a countable set.
  • The probability function, Px, maps each value x in X(Ω) to the probability of the event {X = x}, i.e., Px(x) = P(X = x).
  • The sum of probabilities over all possible values must equal one: Σ Px(x) = 1 where x ∈ X(Ω).
  • The cumulative distribution function (CDF), Fx, is defined as Fx(x) = P(X ≤ x).

Continuous Real Random Variables

  • A continuous RV is a RV whose possible values form an uncountable set.
  • The probability density function (PDF), fx, is a non-negative function such that the probability of X falling within an interval [a, b] is given by the integral of fx over that interval: P(a ≤ X ≤ b) = ∫ab fx(x) dx
  • The cumulative distribution function (CDF), Fx, is defined as Fx(x) = ∫−∞x fx(t) dt

Mode

  • The mode of a discrete RV is the value that maximizes the probability function.
  • For continuous RVs, the mode is the value that maximizes the probability density function.

Percentiles

  • A percentile of order α (0 < α < 1) is a value x such that P(X ≤ x) = α.
  • Specific percentiles include the median (α = 0.5), first quartile (α = 0.25), and third quartile (α = 0.75).

Mathematical Expectation

  • The expectation of a RV X, denoted E(X), is a measure of the central tendency of the distribution.
    • In discrete case: E(X) = Σ x * Px(x)
    • In continuous case: E(X) = ∫ x * fx(x) dx
  • The expectation of a function h(X) is given by E(h(X)) = Σ h(x) * Px(x) for discrete, and E(h(X)) = ∫ h(x) * fx(x) for continuous.

Moments

  • The kth moment of a distribution is defined as E(Xk) for discrete, and ∫ xk * fx(x) for continuous.

  • The kth central moment is E[(X - E(X))k].

  • Variance: The second central moment. E(X-E(X))²

Moment Generating Function (MGF)

  • The MGF, GX(t), is a function that encapsulates important information about the distribution.
  • It can be used to calculate moments of X from its derivatives.

Probability Distribution of a Function of a Real Random Variable

  • The distribution of a function of a RV can be derived using the theory of cumulative distribution functions or probability density functions.

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