My Find the standard deviation of the discret probability dist x P(x) xp(x)) (x-M)² | (x-mipul 2 σ = √(x-M)² pxx) ટે О 0.21 10*0.21=0 /10-112 =. M= Exp(x) 1 0.30 1*0.3 = 0.3 2 0=49 EXP(X) =M
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- Q3 i): A survey of 500 randomly chosen individuals is conducted. The individuals are asked to name their favorite sport. The STATISTIX pie chart in the following figure summarizes the results of this surveyA random variable X has the following pdf: f (x) = 9xe-3";x > 0 %3D (a) Calculate the expected value of the distribution, E(X). (b) Calculate the mode of the distribution (no need to show the second derivative).Find the method of moments estimator for the parameters of the double exponential distribution DE(0,n) f(x) Le-lz-nl/, - 0 0. 20
- The pdf of the time to failure of an electronic component in a copier (in hours) is f (x) = [exp (-x/3100)]/3100 for x > 0 and f (x) = 0 for x ≤ 0. Determine the probability that:(a) A component lasts more than 1180 hours before failure.(b) A component fails in the interval from 1180 to 2010 hours.(c) A component fails before 3010 hours.(d) Determine the number of hours at which 11% of all components have failed.(e) Determine the mean.Round your answers to three decimal places (e.g. 98.765).Let Y be the present value random variable of a 10-year term annuity-due of 1 issued at an individual (35). Calculate its actuarial present value if (a) Mortality follows De Moivre's Law with w = 100. (b) 8 = 0.05.Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
- Suppose that X follows a lognormal distribution with w^2=4 and E(X)=10,000 a. Find the parameter theta for the distribution. Use 4 digits after the decimal point: b. Find the standard deviation of the process: Use 4 digits after the decimal point:For the random variable X with the probability function below, answer the following questions. f(x)= { 3x^2 , 0<x<1 { 0, otherwise a. Find the cdf of X. b. Find the probability that X is less than 0.5. c. Find E(X).The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…
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