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**Linear Functions of Random Variables**

- The random occurrence of defects results in cost of returned items.
- The random variation of stock prices determines the performance of a portfolio.
- The random arrival of patients affects the length of the waiting line in a doctor’s office.

- The random occurrence of defects results in cost of returned items.
- The random variation of stock prices determines the performance of a portfolio.
- The random arrival of patients affects the length of the waiting line in a doctor’s office.

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## 2 5 Linear functions of random variables

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## How do you find the linear functions of random variables?

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Linear functions of random variables. Let X be a random variable and for a, b ∈ R let Y be the random variable. Y ( s) = a ⋅ X ( s) + b. Then, E [ Y] = a ⋅ E [ X] + b Var ( Y) = a 2 ⋅ Var ( X) σ Y = | a | σ X.

**Linear functions** of **random** **variables** Let X be a **random** **variable** and for a, b ∈ R let Y be the **random** **variable** Y ( s) = a ⋅ X ( s) + b Then, E [ Y] = a ⋅ E [ X] + b Var ( Y) = a 2 ⋅ Var ( X) σ Y = | a | σ X. **Linear** combinations of **random** **variables** Let X 1, …, X n be **random** **variables** and c 1, …, c n ∈ R.

How do you find the random variable of a function?

A function X: S → R is called a random variable. Any random variable determines a new probability Q on R. For A ⊂ R we set Q ( A) = P ( X ∈ A). Suppose we roll two fair dice. Let X be the sum of the two rolls. What is S? What is Q ( { 7 }) = P ( X = 7)? A random variable is discrete if its possible values form a discrete (i.e. countable) set.

What is a linear function graph?

Knowing an ordered pair written in function notation is necessary too. f (a) is called a function, where a is an independent variable in which the function is dependent. Linear Function Graph has a straight line whose expression or formula is given by; It has one independent and one dependent variable.

How do you find the variances of independent random variables?

When variables are independent, their variances sum. Let X 1, …, X n be independent random variables. Var ( X 1 + ⋯ + X n) = ∑ i = 1 n Var ( X i).

How do you find the expected value of a linear combination?

For example, if we let X be a random variable with the probability distribution shown below, we can find the linear combination’s expected value as follows: Additionally, this theorem can be applied to finding the expected value and variance of the sum or difference of two or more functions of the random variables X and Y

## What is a linear combination of normal random variables?

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Linear combinations of normal random variables. One property that makes the normal distribution extremely tractable from an analytical viewpoint is its closure under linear combinations: the linear combination of two independent random variables having a normal distribution also has a normal distribution.

**bivariate normal distribution**. It forms the basis for all calculations involving arbitrary means and variances relating to the more general bivariate normal distribution.

What are the linear combinations of normally distributed random variables?

Linear combinations of normally distributed random variables Theory: A. Let X˘ N(;˙). Then the random variable Y = X+ is also normally distributed as follows: Y ˘ N( + ; ˙) B. Let X ˘ N(. X;˙. X) and Y ˘ N(. Y ;˙. Y ). Then, if X and Y are independent, the random variable S= X+ Y follows also the normal distribution with mean .

What is linear combinations?

Linear Combinations is the answer! More importantly, these properties will allow us to deal with expectations (mean) and variances in terms of other parameters and are valid for both discrete and continuous random variables. Let’s quickly review a theorem that helps to set the stage for the remaining properties.

Why do linear combinations have population means and variances?

Because linear combinations are functions of random quantities, they also are random vectors, and hence have population means and variances. Moreover, if you are looking at several linear combinations, they will have covariances and correlations as well. Therefore we are interested in knowing:

How do you find the expected value of a linear combination?

For example, if we let X be a random variable with the probability distribution shown below, we can find the linear combination’s expected value as follows: Additionally, this theorem can be applied to finding the expected value and variance of the sum or difference of two or more functions of the random variables X and Y

## What is a linear function graph?

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Knowing an ordered pair written in function notation is necessary too. f (a) is called a function, where a is an independent variable in which the function is dependent. Linear Function Graph has a straight line whose expression or formula is given by; It has one independent and one dependent variable.

What does the graph of a linear function look like?

We previously saw that that the graph of a linear function is a straight line. We were also able to see the points of the function as well as the initial value from a graph. There are three basic methods of graphing linear functions.

What is a linear function?

A linear function is a function which forms a straight line in a graph. It is generally a polynomial function whose degree is utmost 1 or 0. Although the linear functions are also represented in terms of calculus as well as linear algebra.

What does the graph of the function represent?

The graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant which indicates a negative slope. This is also expected from the negative constant rate of change in the equation for the function.

How do you graph a linear equation?

Graphing a linear equation involves three simple steps: 1 Firstly, we need to find the two points which satisfy the equation, y = px+q. 2 Now plot these points in the graph or X-Y plane. 3 Join the two points in the plane with the help of a straight line. More …

## Which random variable has a normal distribution with mean and variance?

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Then, the random variable defined as: has a normal distribution with mean and variance The sum of more than two independent normal random variables also has a normal distribution, as shown in the following example. Example Let be mutually independent normal random variables, having means and variances .

**random variable X**is said to have a normal probability distribution with parameters μ and σ2, if it has a pdf given by: If μ = 0, and σ = 1, we call it a standard normal random variable. For any normal random variable with mean μ and variance σ2, we use the notation X ∼ N (μ, σ2).

What is variance of a random variable?

Variance of a random variable is discussed in detail here on. Basically, the variance tells us how spread-out the values of X are around the mean value. Variance of a random variable (denoted by σ2 x σ x 2) with values x1,x2,x3,…,xn x 1, x 2, x 3, …, x n occurring with probabilities p1,p2,p3,…,pn p 1, p 2, p 3, …, p n can be given as :

Is x 12 4 a normal random variable?

X is a normal random variable with mean 12 and variance 2 2, meaning that is a normal random variable with mean 0 and variance 1. The probabilities you are interested in can then be found in tables such the ones above. Z = X − 12 4 has variance 1 4. That’s correct, thank you ! Im still unsure how to get any of the probabilities with this though…

What is the standard deviation of a random variable?

Summary A Random Variable is a variable whose possible values are numerical outcomes of a random experiment. The Mean (Expected Value) is: μ = Σxp The Variance is: Var(X) = Σx 2p − μ 2 The Standard Deviation is: σ = √Var(X)

What is the proportion of normal distribution within two standard deviations?

This corresponds to the proportion 0.95 for data within two standard deviations of the mean. which corresponds to the proportion 0.997 for data within three standard deviations of the mean. A standard normal random variable Z is a normally distributed random variable with mean μ = 0 and standard deviation σ = 1.

References:

Linear Functions of Random Variables – Docest

Linear functions of random variables — STATS110 – Stanford …

Linear Functions of Random Variables – Docest

Linear functions of Random Variables – Dr Sandhya Aneja

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What is variance of a random variable?

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What does the graph of a linear function look like?

What is a linear function?

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What is a linear function graph?

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What is a linear function graph?

How do you find the variances of independent random variables?

How do you find the expected value of a linear combination?

How do you find the linear functions of random variables?

What are the linear combinations of normally distributed random variables?

What is linear combinations?

Why do linear combinations have population means and variances?

How do you find the expected value of a linear combination?

What is a linear combination of normal random variables?

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