Q.2h(x)==-exp8You are given h(x) with the random variable X taking only positive values,-7)+X1X1-exp+-expl2X824a. Check if h(x) is a valid pdf (probability density function)?b.Obtain the CDF of X.c. Obtain the mean of X.4-exp80X10,x≥0

The random variable taking only positive values and the pdf is given as follows:

Answered: Q.2 You are e given h(x) with the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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opment of a particular product. For cost reasons, all three are used at varying A manufacturing firm employs three analytical plans for the design and devel. opment of a particular product. For cost reasons, all three are used at varying times. In fact, plans 1, 2, and 3 are used for 30%, 20%, and 50% of the products respectively. The defect rate is different for the three procedures as follows: P(D\P1) = 0.01, P(D\P2) = 0.03, P(D\P3) = 0.02, %3D where P(D|P;) is the probability of a defective product, given plan j. If a random product was observed and found to be defective, which plan was most likely used and thus responsible?
21. (a) Explain why the function TTX sin 10 if 0 10 is a probability density function. (b) Find P(X < 4). (c) Calculate the mean. Is the value what you would expect?
A continuous random variable T (in years) of the lifespan of a device has the following probability density function f (t) = {3 pt", Osts2 otherwise where p is a constant, (a) Determine the value of p. (b) Find the cumulative distribution function of T, F(t). (c) If U =ÉT+1 determine E(U) and Var(U).
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6. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-x, o0) is a function f such that f(r) 20 and (2) = = 1. (a) Determine which of the following functions are probability density functions on the (-x0, 00). fr-1 00 (b) We can also use probability density functions to find the erpected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. rf(x) dr yields the expected value for a density f(r) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.
The normal probability density function g(x) = ov2n Where u = mean, o = standard deviation, n = 3.14159, e 2.71828 Let's study a special case where u = 0 and o = 1, which indicates a standard normal probability density function. For simplicity, we scale the function so as to remove the factor 1/(ov2n). Now, we have a new function f(x) f(x) = e (a) Find f'(x) and show the intervals of increase or decrease for f. (b) Find the extreme value of f. (c) Find the intervals of concavity and the inflection points. (d) Sketch the graph of f and point out the extremum value and inflection points.
do al otherwise not
5. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³
5. Suppose that the random variable X follows an exponential distribution with probability density function S(z) = de-A";0 0). Define a new random variable Y = x. (a) Show that the cumulative density function of Y is given by Fy(y) = {-exp(-Ay"; and hence, or otherwise, find the probability density function of Y. (b) Find an expression for the maximum likelihood estimator of the parameter A, using a sample Y1, 92, , Yn, from the distribution of Y. v20
Just a quick question for a derived random variable, how do you know that f(m) = f(x) abs(dx/dm)? Is that a theorem?

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Q.2 You are e given h(x) with the random variable X taking only positive values, ^(2) - * - ( 4 )- 400 (1) + - (-) - - - ( + ) 1 X X exp X +-expl h(x)==-exp 24 8 8 -exp X 20 4 80 a. Check if h(x) is a valid pdf (probability density function)? b. Obtain the CDF of X. c. Obtain the mean of X.