• Let X be exponential distributed with parameter 1. Determi the probability density function of y = X-3 X + 1
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![1) Let X be exponential distributed with parameter d. Determine
the probability density function of y
X-3
X + 1
2) Let A CR² be the triangle given by (0,0), (1,0), (0, 1). Let
(x, y) be uniformly distributed in A.
a) determine the propability density function of (x, y)
b) SHow that x and y
are dependent
c) the cumulative distribution function of X for OEX EI
is given by fx (x)=2x-x²
1) show by using probability density function
11) show by using direct area determination](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b7636ea-b999-4d69-9ffa-d2ca3c8641a8%2F2011f03b-b4e2-4e86-9c92-7b03f54f3993%2Fn5c0y8i_processed.jpeg&w=3840&q=75)
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- The joint density of X and Y is given by, ху, Iw(x, y) = (x² +: 0 l) b) Find the marginal probability distributions of X and Y. c) Find the conditional probability density function of Y given X=0.5 and calculate P(YLet X₁, X2, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = (0, -, if 0The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.8) Consider the function f(x) = (3x-d)/26 x=1,2,3,4 a) find the constant d, if f(x) is a valid probability density function (pdf)of the random variable X b) find the probability function (list in tabular form) c) find mean and variance of X d) find the moment generating function (mgf)of X e) using the mgf (found in part d) find the mean and standard deviation of XLet X1 and X2 be two continuous random variableshaving the joint probability density f(x1, x2) = 4x1x2 for 0 < x1 < 1, 0 < x2 < 10 elsewhereFind the joint probability density of Y1 = X21 and Y2 = X1X2.The life lengths of two transistors in an electronic circuit is a random vector (X; Y ) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y ) is given by | 2e-(x+2y) x 2 0, y 20 fx,y(x,y) = fx.MX.v) else Then the probability that the first transistor last for at least half hour given that the second one lasts at least half hour equals Select one: a. 0.3669 b. 0.3935 c. 0.7772 d. 0.6318 e. 0.606Let X be a continuous random variable with density function(x) = 4 1. Determine F(5) 2. Determine p(1Let f(x, y) = {siz xy 512 0 a) xy if 0Q1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?Let X be a uniformly distributed continuous random variable that can take values in the interval (-1,1). Y Let Y=1-X2 be defined as a function of the random variable X. a) Mathematically find the probability density function (PDF) of Y. b) Find mathematically the covariance, Cov[X,Y], of X and Y. X and Y are uncorrelated (uncorrelated)? Are X and Y independent? c) With the help of the "rand" command in MATLAB, the random variable X is randomized into 1,000,000 random variables. and the corresponding Y=1-X2 values. Plot the histogram of Y. The histogram you found in (a) Does it match the PDF? Comment. d) Estimate Cov[X,Y] using the 1,000,000 X and Y values you generated in MATLAB obtain For this, from the definition Cov[X,Y] = E[(X-E[X])(Y-E[Y])] and the "mean" command Does it match the value you found in (b)? Comment.Determine the value of c that makes the function f(x, y) = c(x + y) a joint probability mass function over the nine points with x = 1,2,3 and y = 1,2,3. Give exact answer in form of fraction.The life lengths of two transistors in an electronic circuit is a random vector (X; Y ) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y ) is given by ´2e-(x+2) x 2 0, y 20 fx,y(x,y)=| else Then the probability that the first transistor burned during half hour given that the second one lasts at least half hour equals Select one: a. 0.3935 b. 0.606 c. 0.7772 d. 0.3669 e. 0.6318SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. 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