1. Answer the following questions. (a) Check whether or not each of f1(x), f2(x) is a legitimate probability density function f1(x) = ( 0.5(3x − x3) 0< x < 2 0 otherwise f2(x) = ( 0.5(3x − x2) 0< x < 2 0 otherwise (b) Let X denote the resistance of a randomly chosen resistor, and suppose that its PDF is given by f(x) = ( kx 8  x  10 0 otherwise (i) Find k and the CDF of X, and use the CDF to calculate P(8.6  X  9.8). (ii) Find the conditional probability that X  9.8 given that X # 8.6.

f1 ( x ) valid pdf . f2 ( x ) is invalid pdf

k = 1 / 18 , i ) 0.6133 , ii ) 0.84792

Step-by-step explanation:

Solution:-

A) The two pdfs ( f1 ( x ) and f2 ( x ) ) are given as follows:

- To check the legitimacy of a continuous probability density function the area under the curve over the domain must be equal to 1. In other words the following:

- We will perform integration of each given pdf as follows:

f1 ( x ) is a valid pdf; however, f2 ( x ) is not a valid pdf.

B)

- A random variable ( X ) denotes the resistance of a randomly chosen resistor, and the pdf is given as follows:

                        if  8 ≤ x ≤ 10

                                0  otherwise.

- To determine the value of ( k ) we will impose the condition of validity of a probability function as follows:

- Evaluate the integral as follows:

                      ... Answer

- To determine the CDF of the given probability distribution we will integrate the pdf from the initial point ( 8 ) to a respective value ( x ) as follows:

To determine the probability p ( 8.6 ≤ x ≤ 9.8 ) we will utilize the cdf as follows:

                    p ( 8.6 ≤ x ≤ 9.8 ) = F ( 9.8 ) - F ( 8.6 )

                    p ( 8.6 ≤ x ≤ 9.8 ) =

ii) To determine the conditional probability we will utilize the basic formula as follows:

                p ( x ≤ 9.8  | x ≥ 8.6 ) = p ( 8.6 ≤ x ≤ 9.8 ) / p ( x ≥ 8.6 )

                p ( x ≤ 9.8  | x ≥ 8.6 ) = 0.61333 / [ 1 - p ( x ≤ 8.6 ) ]

                p ( x ≤ 9.8  | x ≥ 8.6 ) = 0.61333 / [ 1 - 0.27666 ]

                p ( x ≤ 9.8  | x ≥ 8.6 ) = 0.61333 / [ 0.72333 ]

                p ( x ≤ 9.8  | x ≥ 8.6 ) = 0.84792 ... answer

6mm^3 or .6cm^3

      216

      /   \

  36     6

  /   \     /

6     6   6

factor trees are nice.