3. Problem 3 This problem involves a battery-operated critical medical monitor. The lifetime (T) of the battery is a random variable with an exponentially distributed lifetime. A battery lasts an average of 45 days. Under these conditions, the PDF of the battery lifetime is given by: (ae-at, t0 fr(t)= 0, where a = 1 t 3x365)-1-P(S3x365)-1-F(1095). You can use the graph of the CDF F(r) you created previously, to estimate this probability. 2. Find the probability that the carton will last between 2.0 and 2.5 years (ie. between 730 and 912 days): P(730 S912) F(912) - F(730). You can use the graph of the CDF F(t) you created previously, to estimate this probability. SUBMIT the following: a Word file with the numerical answers using the format in the table below the PDF plot of the lifetime of one carton the CFD plot of the lifetime of one carton the MATLAB code. Make sure that the graphs are properly labeled. QUESTION 1. Probability that the carton will last longer than three years 2. Probability that the carton will last between 2.0 and 2.5 years Ans. 0.3. Simulate a R.V. with Arbitrary Probability Distribution As mentioned above, the standard MATLAB function "rand" will generate random variables with uniform probability distribution in the interval [0,1]. In many situations, however, you need to generate RVs with different distributions, such as the Poisson distribution. This section will provide you with the tools to generate RVs with arbitrary probability distributions using the uniform RV generate by the function "rand". To see how this can be done consider the following: Let X be a continuous random variable with an arbitrary cumulative distribution function F(x). Assume for our purposes that F(x) is strictly increasing, so that the inverse function F-(y) exists for all y. Let U be a new random variable, defined as: U = F(x). Obviously, the range of I will be in the interval [0, 1], since I is equal to a CDF. Then: P(U 0, x-a u = Fx(x)= b 1, x 2. Problem 2 Exponentially Distributed RVs Simulate an exponentially distributed RV (T) based on the discussion of a previous section. The generated RV T should have the following PDF fr(t)= 0, t