How to find the probability using discrete variables?

Question

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

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Answer 1 · 2018-01-13T12:23:13

#(a)" "k=1/21#

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

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

#(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#