How to find the probability using discrete variables?

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Can someone please explain to me how to do questions 10? Thanks heaps!

1 Answer
Jan 13, 2018

(a)" "k=1/21

(b)" "E(X)=13/3

(c)" "Var(X)=20/9

Explanation:

(a)" "P(X=r)=kr" for "r=1,2,3,4,5,6

now

sum_"all r"P(X=r)=sum_(r=1)^6kr=1

substitute for the values of r

:.k+2k+3k+4k+5k+6k=1

21k=1

=>k=1/21

(b)" "E(X)

E(X)=sum_("all" "r)r P(X=r)=sumrkr=ksumr^2

:.E(X)=1/21(1+2xx2+3xx3+4xx4+5xx5+6xx6)

=1/21(1+4+9+16+25+36)

=91/21=13/3

(c)" "Var(X)

Var(X)=E(X^2)-E^"(X)

E(X^2)=sumr^2P(X=r)=sumr^2kr=1/21sumr^3

=1/21(1+8+27+64+125+216)=441/21=21

:.Var(X)=21-(13/3)^2=21-169/9=20/9