A certain type o component is packaged in lots of four. Let X represent the numb

Question

A certain type o component is packaged in lots of four. Let X represent the number of properly functioning components in arandomly chosen lot. Assume that the probability that exactlyx components function is proportional to x; in other words, assumethat the probability mass function of X is given by

   P(x){cx   1 , 2 , 3 or 4

       0 otherwise

Where c is a constant

A)    Find the value of the constant c so that p(x)is a probability mass function.

B)    Find p(X = 2)

C)    Find the mean number of properly functioningcomponents.

D)    Find the variance of the number of properlyfunctioning components.

E)     Find the standard deviation of thenumber of properly functioning components.

Explanation / Answer

P(x){cx   1 , 2 , 3 or 4

A)    Find the value of the constant c so that p(x)is a probability mass function.

        c(1+2+3+4)=1 -- > c=1/10=0.1

B)    Find p(X = 2)

        P(X=2)=0.1*2=0.2

C)    Find the mean number of properly functioningcomponents.

       mean=E(X)=1*0.1+2*0.2+3*0.3+4*0.4=3

D)    Find the variance of the number of properlyfunctioning components.

      E(X^2)=1*0.1+2^2*0.2+3^2*0.3+4^2*0.4=10

       Var(X)=E(X^2)-[E(X)]^2=10-3^2=10-9=1

E)     Find the standard deviation of thenumber of properly functioning components.

        Sd(X)=Var(X)=1=1

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