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Questions and Answers
What is the integral used to calculate the expected value E(X)?
What is the integral used to calculate the expected value E(X)?
What is the expected value E(4X + 3) if E(X) = 5/4?
What is the expected value E(4X + 3) if E(X) = 5/4?
Which of the following values is the result of $\int_{-\infty}^{\infty} x^3 dx$?
Which of the following values is the result of $\int_{-\infty}^{\infty} x^3 dx$?
What is the value of E(X) calculated from the provided integrals?
What is the value of E(X) calculated from the provided integrals?
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If f(x) is equal to x for the range given, which statement is true about E(X)?
If f(x) is equal to x for the range given, which statement is true about E(X)?
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What is the formula to find the variance of a linear combination of independent random variables?
What is the formula to find the variance of a linear combination of independent random variables?
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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?
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?
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How does the constant term affect the variance of the random variable Z = 3X - 4Y + 8?
How does the constant term affect the variance of the random variable Z = 3X - 4Y + 8?
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If the variances of X and Y are combined in the expression Z = 3X - 4Y + 8, what is the variance of Z?
If the variances of X and Y are combined in the expression Z = 3X - 4Y + 8, what is the variance of Z?
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Which of the following statements about independent random variables X and Y is true when calculating the variance of Z?
Which of the following statements about independent random variables X and Y is true when calculating the variance of Z?
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What is the formula for the variance of a linear combination of independent random variables?
What is the formula for the variance of a linear combination of independent random variables?
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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)?
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)?
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When combining two random variables with their respective coefficients, which relationship holds true for their variance?
When combining two random variables with their respective coefficients, which relationship holds true for their variance?
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Which of the following statements regarding the variance of linear combinations is FALSE?
Which of the following statements regarding the variance of linear combinations is FALSE?
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Given independent random variables X and Y, which expression for their combined variance correctly utilizes the coefficients a and b?
Given independent random variables X and Y, which expression for their combined variance correctly utilizes the coefficients a and b?
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What is the variance of the expression $3X - 4Y + 8$?
What is the variance of the expression $3X - 4Y + 8$?
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In the expression $3X - 4Y + 8$, which term contributes zero to the variance?
In the expression $3X - 4Y + 8$, which term contributes zero to the variance?
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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?
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?
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Which formula is used to determine the variance of the expression $3X - 4Y + 8$?
Which formula is used to determine the variance of the expression $3X - 4Y + 8$?
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What is the contribution of the term $-4Y$ to the overall variance in the expression?
What is the contribution of the term $-4Y$ to the overall variance in the expression?
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What is the expected value of a linear transformation of a random variable, if the transformation is given by setting 𝑏 = 0?
What is the expected value of a linear transformation of a random variable, if the transformation is given by setting 𝑏 = 0?
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Given the probability density function of a random variable 𝑋, what is the range of 𝑓(x) where it is non-zero?
Given the probability density function of a random variable 𝑋, what is the range of 𝑓(x) where it is non-zero?
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What does the probability density function 𝑓(𝑥) represent in this case?
What does the probability density function 𝑓(𝑥) represent in this case?
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If you compute 𝐸(4𝑋 + 3), what is the multiplicative factor applied to the expected value of 𝑋?
If you compute 𝐸(4𝑋 + 3), what is the multiplicative factor applied to the expected value of 𝑋?
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When calculating 𝐸(4𝑋 + 3), which operation must be performed first based on the linearity of expectation?
When calculating 𝐸(4𝑋 + 3), which operation must be performed first based on the linearity of expectation?
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What is the expected value of the expression $E(3X + 7)$?
What is the expected value of the expression $E(3X + 7)$?
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Which expression correctly represents the variance of a linear combination $Var(5X + 2Y - 2)$ under the assumption that X and Y are independent?
Which expression correctly represents the variance of a linear combination $Var(5X + 2Y - 2)$ under the assumption that X and Y are independent?
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What impact does adding a constant, such as -2 in $Var(5X + 2Y - 2)$, have on the variance?
What impact does adding a constant, such as -2 in $Var(5X + 2Y - 2)$, have on the variance?
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If $X$ has an expected value of 4 and $Y$ has an expected value of 3, what is the expected value of the expression $E(5X + 2Y - 2)$?
If $X$ has an expected value of 4 and $Y$ has an expected value of 3, what is the expected value of the expression $E(5X + 2Y - 2)$?
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What is the relationship between the variance of the sum of independent random variables $Var(X + Y)$ and their individual variances?
What is the relationship between the variance of the sum of independent random variables $Var(X + Y)$ and their individual variances?
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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
Studying That Suits You
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Description
Test your understanding of linear combinations of random variables with this quiz. The focus is on Theorems and Corollaries related to the expected value of random variables. Challenge yourself with questions that require applying key concepts from probability and statistics for engineers.
