functions and transformations of random variables
DESCRIPTION
Section 09. Functions and Transformations of Random Variables. Transformation of continuous X. Suppose X is a continuous random variable with pdf and cdf Suppose is a one-to-one function with inverse ; so that The random variable is a transformation of X with pdf: - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/1.jpg)
Functions and Transformations of Random Variables
Section 09
![Page 2: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/2.jpg)
Transformation of continuous X Suppose X is a continuous random variable
with pdf and cdf Suppose is a one-to-one function with
inverse ; so that
The random variable is a transformation of X with pdf:
If is a strictly increasing function, then and then
![Page 3: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/3.jpg)
Transformation of discrete X
Again, Since X is discrete, Y is also discrete
with pdf
This is the sum of all the probabilities where u(x) is equal to a specified value of y
![Page 4: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/4.jpg)
Transformation of jointly distributed X and Y
X and Y are jointly distributed with pdf and are functions of x and y This makes and also random variables
with a joint distribution
In order to find the joint pdf of U and V, call it g(u,v), we expand the one variable case Find inverse functions and so that and Then the joint pdf is:
![Page 5: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/5.jpg)
Sum of random variables
If then
If Xs are continuous with joint pdf
If Xs are discrete with joint pdf
![Page 6: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/6.jpg)
Convolution method for sums
If X1 and X2 are independent, we use the convolution method for both discrete & cont.
Discrete:
Continuous:
![Page 7: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/7.jpg)
Sums of random variables
If X1, X2, …, Xn are random variables and
If Xs are mutually independent
![Page 8: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/8.jpg)
Central Limit Theorem
X1, X2, …, Xn are independent random variables with the same distribution of mean μ and standard deviation σ
As n increases, Yn approaches the normal distribution
Questions asking about probabilities for large sums of independent random variables are often asking to use the normal approximation (integer correction sometimes necessary).
![Page 9: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/9.jpg)
Sums of certain distribution
This table is on page 280 of the Actex manualDistribution of Xi Distribution of Y
Bernoulli B(1,p) Binomial B(k,p)
Binomial B(n,p) Binomial B( ,p)
Poisson Poisson
Geometric p Negative binomial k,p
Normal N(μ,σ2) Normal N( ,)
There are more than these but these are the most common/easy to remember
![Page 10: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/10.jpg)
Distribution of max or min of random variables
X1 and X2 are independent random variables
![Page 11: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/11.jpg)
Mixtures of Distributions
X1 and X2 are independent random variables We can define a brand new random variable X as
a mixture of these variables! X has the pdf
Expectations, probabilities, and moments follow this “weighted-average” form
Be careful! Variances do not follow weighted-average! Instead, find first and second moments of X and subtract
![Page 12: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/12.jpg)
Sample Exam #101
The profit for a new product is given by Z = 3X – Y – 5 . X and Y are independent random variables with Var(X)=1 and Var(Y)=2.
What is the variance of Z?
A) 1 B) 5 C) 7 D) 11 E) 16
![Page 13: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/13.jpg)
Sample Exam #102
A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails.
What is the variance of the total time that the generators produce electricity?
![Page 14: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/14.jpg)
Sample Exam #103
In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4
Determine the probability that the maximum of these losses exceeds 3.
![Page 15: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/15.jpg)
Sample Exam #142
An auto insurance company is implementing a new bonus system. In each month, if a policyholder does not have an accident, he or she will receive a $5 cash-back bonus from the insurer.
Among the 1000 policyholders of the auto insurance company, 400 are classified as low-risk drivers and 600 are classified as high-risk drivers.
In each month, the probability of zero accidents for high-risk drivers is .8 and the probability of zero accidents for low-risk drivers is .9
Calculate the expected bonus payment from the insurer to the 1000 policyholders in one year.
![Page 16: Functions and Transformations of Random Variables](https://reader036.vdocument.in/reader036/viewer/2022062321/56813056550346895d960e40/html5/thumbnails/16.jpg)
Sample Exam #123
You are given the following information about N, the annual number of claims for a randomly selected insured:
Let S denote the total annual claim for an insured. When N=1, S is exponentially distributed with mean 5. When N>1, S is exponentially distributed with mean 8.
Determine P(4<S<8).