Show that if X and Y are two discrete random variables (i.e., elementary random variables) then ZX + Y is also a discrete random variable.
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- 3) A random variable X takes the values 0, 2 and 3 with probabilities 0.3, 0.1 and 0.6, respectively. A random variable in the form of Y = 3(X-1)2 is defined. a) Find the expected value and variance of the random variable X. b) Find the variance of the Y random variable. c) Find the cumulative distribution function of the Y random variable.Let X1, X2,..., X3 denote a random sample from a population having mean u and variance o?... Which of the estimators have a variance of 7 X1+X2++X, 7 2X1-X6+X4 2 3X1-X3+X4 2 2(X1+X2+.+X¬) 4 7If a variable can take certain integer values between two given points, then it is called O a. Continuous random variable O b. Probabilistic random variable c. O c. Deterministic random variable O d. Uncertain random variable O e. Discrete random variable
- 2. Some properties of Expected value and variance of a random variable. a) Assume that X is an arbitrary discrete random variable, and a and b are constant. Using the definitic Show: and E(aX + b) = a · E(X) + b V(aX + b) = a² · V(X ) Stat 3128 Ali Mahzarnia P STAT 3128 Ali Mahzarnia b) Justify the computational formula of Variance of a random variable which is to justify : V(X) = E[(X – µ°] = Ex – µ)P • ptx) = | 2: Here needs justification By Cauchy Schwarz inequality it can be shown that the right hand side is always positive. Analogs expression in Mechanic : Parallel axis theorem Iem = I– md² Moment of Moment of inetria about Inertia of an an axis shifted object about the center of a mass by d from center of mass (a parallel shift) Icm is dispersion around the mean and is like second central moment (variance) I is like second moment if d is mean m is like sum total all the weight of each of the x which all add up to 1 d squared is like squared of mean since we,ag I contains 2 White chips and 1 red chip and Bag II contains 2 red chips and one white chip. A chip is selected at random from g I and transferred to Bag II. Then a chip is selected at random from Bag II. If the selected chip is red, what is the probability that e transferred chip was white? 317 B 2/3 O 1/3 4/7Let X,, X2, and X3 be independent random variables, each are binomially distributed with n = 100 and p = 0.2. Let A = X,– 2X2 and B = X3 + 3X1. Find PAB- %3D %3D
- 2. We have two fair dice, one red and one blue. When we roll them together, the outcome can be shown as an order pair, (R, B) where R and B are numbers from the red and the blue die, respectively. Let X be a random variable defined by X(R, B) = R - B where R and B are numbers from red and blue dice, respectively. (a) What is the probability mass function for the random variable? Show that as a table.prove that if random variables X and Y have same characteristic function, then have same distribution.2 A random sample of size 12 drawn from a population of size 200. The sample values and their probabilities are (52,0.004), (72,0.003), (64,0.005), (58,0.003), (42,0.002), (76,0.006), (71,0.006), (63,0.006), (87,0.008), (57,0.002), (61,0.003) and (86,0.007). Estimate population total with a 95% confidence interval by using probability proportional to size sampling.
- Do question 2Given the joint probabilities of two discrete random variables (X, Y) as below.f(1,1)=4/20, f(1,2)= 3/20, f(1,3)= 1/20, f(2,1)= 2/20, f(2,2)=1/20, f(2,3)= 2/20, f(3,1)= 2/20, f(3,1)=3/20, f(3,3)= 2/20 . Find. i) The mean of a random variable Xii) The variance of random variable X.A university computer laboratory installs one of three operating systems on each computer used in the lab. It is known from product testing that during an hour of web browsing the probability a computer with system 1 installed will crash is 0.15, the probability of a computer with operating system 2 crashing is 0.08 and the probability of a computer with system 3 crashing is 0.1. Operating systems 1 and 3 are installed on the same number of computers while system 2 is installed on twice as many computers as system 1 (or system 3).a) What is the probability that a randomly selected computer crashes during an hour of web browsing?b) If a computer selected at random crashed during an hour of browsing the web, what is the probability it has operating system 2 installed?c) If a computer selected at random does not crash during an hour of web browsing what is the probability it has operating system 1 or operating system 2 installed?
