STAT 301 Linear Combinations of Random Variables

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Questions and Answers

What is the integral used to calculate the expected value E(X)?

  • $\int_{-1}^{2} x^2 dx$
  • $\int_{-\infty}^{\infty} x f(x) dx$ (correct)
  • $\int_{-1}^{3} x^3 dx$
  • $\int_{0}^{3} x^4 dx$

What is the expected value E(4X + 3) if E(X) = 5/4?

  • 7
  • 5
  • 10
  • 8 (correct)

Which of the following values is the result of $\int_{-\infty}^{\infty} x^3 dx$?

  • 0 (correct)
  • $\infty$
  • Undefined
  • Non-zero finite number

What is the value of E(X) calculated from the provided integrals?

<p>5/4 (C)</p> Signup and view all the answers

If f(x) is equal to x for the range given, which statement is true about E(X)?

<p>E(X) is greater than 1 (D)</p> Signup and view all the answers

What is the formula to find the variance of a linear combination of independent random variables?

<p>Var(aX + bY) = a^2Var(X) + b^2Var(Y) (B)</p> Signup and view all the answers

Given the random variables X and Y with variances $\sigma^2_X = 2$ and $\sigma^2_Y = 4$, what is the contribution of the variable Y to the variance of Z = 3X - 4Y + 8?

<p>16 (D)</p> Signup and view all the answers

How does the constant term affect the variance of the random variable Z = 3X - 4Y + 8?

<p>It has no effect on the variance. (B)</p> Signup and view all the answers

If the variances of X and Y are combined in the expression Z = 3X - 4Y + 8, what is the variance of Z?

<p>15 (C)</p> Signup and view all the answers

Which of the following statements about independent random variables X and Y is true when calculating the variance of Z?

<p>The variances are multiplied by their respective coefficients squared. (D)</p> Signup and view all the answers

What is the formula for the variance of a linear combination of independent random variables?

<p>Var(aX - bY) = a^2Var(X) + b^2Var(Y) (B), Var(aX + bY) = a^2Var(X) + b^2Var(Y) (D)</p> Signup and view all the answers

If X and Y are independent random variables and you have Var(aX + bY) = 25, which of the following could be a possible value for Var(X)?

<p>a = 3, b = 4, Var(X) = 16 (C), a = 4, b = 1, Var(X) = 6.25 (D)</p> Signup and view all the answers

When combining two random variables with their respective coefficients, which relationship holds true for their variance?

<p>Var(aX + bY) = a^2Var(X) + b^2Var(Y) (A), Var(aX ± bY) = a^2Var(X) + b^2Var(Y) (D)</p> Signup and view all the answers

Which of the following statements regarding the variance of linear combinations is FALSE?

<p>Independent variables have no impact on each other's variance calculations. (A)</p> Signup and view all the answers

Given independent random variables X and Y, which expression for their combined variance correctly utilizes the coefficients a and b?

<p>Var(aX ± bY) = a^2Var(X) ± b^2Var(Y) (A), Var(aX ± bY) = aVar(X) + bVar(Y) (D)</p> Signup and view all the answers

What is the variance of the expression $3X - 4Y + 8$?

<p>82 (A)</p> Signup and view all the answers

In the expression $3X - 4Y + 8$, which term contributes zero to the variance?

<p>$8$ (C)</p> Signup and view all the answers

If the variance of $X$ is 2 and the variance of $Y$ is 4, what is the coefficient of variance contributed by $X$ in the expression?

<p>9 (B)</p> Signup and view all the answers

Which formula is used to determine the variance of the expression $3X - 4Y + 8$?

<p>$Var(aX + bY + c) = a^2Var(X) + b^2Var(Y)$ (D)</p> Signup and view all the answers

What is the contribution of the term $-4Y$ to the overall variance in the expression?

<p>16 (B)</p> Signup and view all the answers

What is the expected value of a linear transformation of a random variable, if the transformation is given by setting 𝑏 = 0?

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

Given the probability density function of a random variable 𝑋, what is the range of 𝑓(x) where it is non-zero?

<p>−1 &lt; 𝑥 &lt; 2 (B)</p> Signup and view all the answers

What does the probability density function 𝑓(𝑥) represent in this case?

<p>It indicates the likelihood of 𝑋 taking on any value within its range. (D)</p> Signup and view all the answers

If you compute 𝐸(4𝑋 + 3), what is the multiplicative factor applied to the expected value of 𝑋?

<p>4 (C)</p> Signup and view all the answers

When calculating 𝐸(4𝑋 + 3), which operation must be performed first based on the linearity of expectation?

<p>Both operations can be done simultaneously. (D)</p> Signup and view all the answers

What is the expected value of the expression $E(3X + 7)$?

<p>$3E(X) + 7$ (C)</p> Signup and view all the answers

Which expression correctly represents the variance of a linear combination $Var(5X + 2Y - 2)$ under the assumption that X and Y are independent?

<p>$25Var(X) + 4Var(Y)$ (B)</p> Signup and view all the answers

What impact does adding a constant, such as -2 in $Var(5X + 2Y - 2)$, have on the variance?

<p>It does not affect the variance. (A)</p> Signup and view all the answers

If $X$ has an expected value of 4 and $Y$ has an expected value of 3, what is the expected value of the expression $E(5X + 2Y - 2)$?

<p>31 (C)</p> Signup and view all the answers

What is the relationship between the variance of the sum of independent random variables $Var(X + Y)$ and their individual variances?

<p>$Var(X + Y) = Var(X) + Var(Y)$ (A)</p> Signup and view all the answers

Flashcards

E(aX) = aE(X)

The expected value of a constant multiplied by a random variable is equal to the constant multiplied by the expected value of the random variable.

Linear Combination of Random Variables

A linear combination of random variables is an expression of the form aX+bY+ ... , where a, b, ... are constants and X, Y, ... are random variables.

Probability Density Function (PDF)

The probability density function defines the probability of a continuous random variable taking on a particular value.

Expected Value of a Linear Combination

To find the expected value of a linear combination of random variables, we can simply calculate the expected value of each term individually and then add them up.

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Expected Value (E(X))

The expected value of a random variable represents the average value we expect the variable to take on over many trials.

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Linearity of Expectation

The expected value of a linear combination of random variables is equal to the linear combination of the expected values of those random variables.

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Expected Value of a Random Variable

The expected value of a random variable X, denoted as E(X), is the average value of X over its entire probability distribution.

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Random Variable

In the context of statistics, a random variable is a variable whose value is a numerical outcome of a random phenomenon.

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Expectation of a Constant Multiple

The expectation of a constant multiplied by a random variable is equal to the constant multiplied by the expectation of the random variable.

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Variance of a Linear Combination

The variance of a linear combination of random variables is equal to the sum of the variances of each term, multiplied by the square of their respective coefficients.

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Variance of a Constant

The variance of a constant is always zero. This is because a constant value does not vary.

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Variance of a Constant Multiple

The variance of a random variable multiplied by a constant is equal to the square of the constant times the original variance.

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Variance of a Random Variable

The variance of a random variable is a measure of how spread out its values are, on average, around its mean.

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Variance of Independent Variables

If X and Y are independent random variables, the variance of their linear combination (aX + bY) is simply the sum of the variances of each variable, multiplied by the squares of their respective coefficients.

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General Variance of a Linear Combination

The variance of a linear combination of random variables is the sum of the variances of each individual variable multiplied by the square of its corresponding coefficient plus twice the sum of the covariances between each pair of variables, multiplied by their respective coefficients.

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Variance of a Scaled Variable

The variance of a random variable (X) multiplied by a constant (a) is equal to the square of the constant multiplied by the variance of the variable.

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Variance of a Sum of Variables

The variance of a sum of random variables is equal to the sum of the variances of the individual variables plus twice the sum of their covariances.

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Variance of aX

The variance of a constant multiplied by a random variable is equal to the square of the constant multiplied by the variance of the random variable.

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Variance of Z = 3X - 4Y + 8

If X and Y are independent random variables, then the variance of aZ, which is a linear combination of X and Y, can be calculated using the formula: Var(aX + bY + c) = a^2 * Var(X) + b^2 * Var(Y).

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Variance (σ²)

The variance measures the spread or dispersion of a random variable around its mean. It represents the average squared deviation from the mean.

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Independence of Random Variables

Two random variables are independent if the outcome of one does not affect the outcome of the other. In other words, they are not related.

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E(aX + bY)

The expected value of a linear combination of random variables is the sum of the expected values of each individual term.

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Var(aX + bY)

The variance of a linear combination of random variables is calculated by squaring the coefficients, multiplying by the variance of each variable, and adding the results.

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Var(aX) = a²Var(X)

The variance of a constant multiplied by a random variable is equal to the square of the constant multiplied by the variance of the random variable.

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Var(aX + bY) when X & Y are independent

The variance of a linear combination of independent random variables is the sum of the variances of each term, multiplied by the square of their respective coefficients.

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Study Notes

Course Information

  • Course title: Probability and Statistics for Engineers
  • Course code: STAT 301& 305
  • Semester: First Semester 1445 H
  • Department: Mathematics
  • University: Taibah University
  • Faculty: Science

Linear Combinations of Random Variables

  • A linear combination of random variables is a sum of the form: Y = Σ aᵢ Xᵢ = a₁X₁ + a₂X₂ + ... + aₙXₙ, where X₁, X₂, ..., Xₙ are random variables and a₁, a₂, ..., aₙ are constants.

Theorem 1

  • If X is a random variable with mean μₓ = E(X), and if a and b are constants, then:
    • E(aX + b) = aE(X) + b
    • μₐₓ₊ₓ = aμₓ ± b

Corollary 1

  • If a = 0 in Theorem 1, then E(b) = b

Corollary 2

  • If b = 0 in Theorem 1, then E(aX) = aE(X)

Theorem 2

  • If X₁, X₂, ..., Xₙ are n random variables and a₁, a₂, ..., aₙ are constants, then:
    • E(a₁X₁ + a₂X₂ + ... + aₙXₙ) = a₁E(X₁) + a₂E(X₂) + ... + aₙE(Xₙ)
    • E(Σ aᵢ Xᵢ) = Σ aᵢ E(Xᵢ)

Corollary

  • If X and Y are random variables, then E(X ± Y) = E(X) ± E(Y)

Theorem 3

  • If X is a random variable with variance Var(X) = σ²ₓ and if a and b are constants, then:
    • Var(aX ± b) = a²Var(X)
    • σ²ₐₓ₊ₓ = a²σ²ₓ

Corollary 1

  • If a = 1 in Theorem 3, then σ²ₓ₊ₓ = σ²ₓ

Corollary 2

  • If b = 0 in Theorem 3, then σ²ₐₓ = a²σ²ₓ

Theorem 4

  • If X₁, X₂, ..., Xₙ are n independent random variables and a₁, a₂, ..., aₙ are constants, then:
    • Var(a₁X₁ + a₂X₂ + ... + aₙXₙ) = a₁²Var(X₁) + a₂²Var(X₂) + ... + aₙ²Var(Xₙ)
    • Var(Σ aᵢ Xᵢ) = Σ aᵢ²Var(Xᵢ)

Corollary

  • If X and Y are independent random variables, then:
    • Var(aX ± bY) = a²Var(X) + b²Var(Y)

Example 1 (Linear Combination of Random Variables)

  • Find E(4X + 3) given a probability density function of X
  • Show solution with calculations.

Example 2 (Linear Combination of Random Variables)

  • Find the variance of Z = 3X – 4Y + 8, given X and Y are independent, σ²ₓ = 2, σ²ᵧ = 4
  • Show solution with calculations

Example 3

  • Find E(3x + 7) and Var(3x + 7) given μₓ = 2, σ²ₓ = 4, μᵧ = 7, σ²ᵧ = 1
  • Find E(5x + 2y – 2) and Var(5x + 2y – 2)
  • Show solutions with calculations

Example 4 (Linear Combination of Random Variables)

  • Find E(XY and 2X – 3Y), given μₓ = 2, and μᵧ = 7 where X and Y are independent
  • Show solution with calculations

Exercise 1 (Discrete Random Variable)

  • Find the cumulative distribution function, expected value, variance and more for a discrete random variable X given a probability mass function.
  • Example Exercise Q1: Given a probability distribution table for X.

Exercise 2 (Continuous Random Variable)

  • Verify that a function f(x) is a probability density function for a continuous random variable X defined for interval 0 ≤ x ≤ 4
  • Find P(1 ≤ X ≤ 3) and others
  • Find the cumulative distribution function, expected value and variance of a continuous random variable.
  • Example Exercise Q2: Given a function f(x) defined for interval 0 ≤ x ≤ 4.

Exercise 3 (Continuous Random Variable)

  • Find the value of c and P(X > 1/2) given a probability density function for a continuous random variable X for 0 ≤ x ≤ 1.
  • Example Exercise Q3:

Exercise 4 (Cumulative Distribution Function)

  • Find f(x) given a cumulative distribution function F(x).
  • Show solution with calculations.
  • Example Exercise Q4:Given a cumulative distribution function

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