Question

suppose x is a continuous random variable with the probability density function given f(x)={Kx ,2k,6k-kx 0<=x<=2, 2<=x<=4,4<=x<=6 find the value of k, cumulative distribution function ,mean,variance and standard deviation ​

Answer 1

Given : suppose x is a continuous random variable with the probability density function given f(x)={Kx ,2k,6k-kx 0<=x<=2, 2<=x<=4,4<=x<=6

To Find : value of k

Solution:

f(x) =  Kx    0<=x<=2

        2k,    2<=x<=4

        ,6k-kx  4<=x<=6

$\int\limits^2_0 {Kx} \, dx + \int\limits^4_2 {2k} \, dx + \int\limits^6_4 {(6k-kx)} \, dx$

= [Kx²/2 ] $_{0} ^2$  + [2kx ] $^4_{2}$  + [6kx -kx²/2 ] $^6_{4}$  

= 2k - 0 + 8k - 4k + (36k - 36k/2 -(24k - 16k/2))

= 2k + 4k + 2k

= 8k

8k = 1

=> k = 1/8

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