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Theorem: 4.1
If X and Y be two symmetric trapezoidal fuzzy random variables with means “?” _”1″ “,” “?” _”2″ and standard deviations “?” _”1″ “,” “?” _”2″ respectively, then the membership function of multiplication of two symmetric trapezoidal fuzzy random variables is
?”P{((X?Y)” ?_”?” ^”L” – “?” _”1″ “?” _”2″ (“?” /”2″ –”1″ )–”?” _”2″ “?” _”1″ (“?” /”2″ –”1″ )” – ” “?” _”1″ “?” _”2″ (“1 – ” “3?” /”4″ ) –”?” _”1″ “?” _”2″ “) ? 0”
“? ” ?”((X?Y)” ?_”?” ^”U” ” + ” “?” _”1″ “?” _”2″ (“?” /”2″ –”1″ )” + ” “?” _”2″ “?” _”1″ (“?” /”2″ –”1″ )” + ” “?” _”1″ “?” _”2″ (“3?” /”4″ –”1″ )” – ” “?” _”1″ “?” _”2″ “) ? 0}”

Proof:
The membership function of symmetric trapezoidal fuzzy random variables X and Y with means “?” _”1″ “,” “?” _”2″ and standard deviations “?” _”1″ “,” “?” _”2″ respectively is
?”X = P{(X” ?_”?” ^”L” ” + (1″-“?” /”2″ “)” “?” _”1″ -“?” _”1″ “) ? 0 ? (” “X” _”?” ^”U” -“(1”-“?” /”2″ “)” “?” _”1″ -“?” _”1″ “) ? 0}”
?”Y = P{(Y” ?_”?” ^”L” ” + (1″-“?” /”2″ “)” “?” _”2″ -“?” _”2″ “) ? 0 ? (” “Y” _”?” ^”U” -“(1”-“?” /”2″ “)” “?” _”2″ -“?” _”2″ ” ? 0},”
According to Vahidi. J, Rezvani. S 7, the coordinate points of multiplication of two symmetrical trapezoidal fuzzy random variables is
(“(” “?” _”1″ –”?” _”1″ “)(” “?” _”2″ – “?” _”2″ “),(” “?” _”1″ – “?” _”1″ /”2″ “)(” “?” _”2″ – “?” _”2″ /”2″ “),(” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “),(” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ –”?” _”2″ “),” )
then the membership function of multiplication of two symmetric trapezoidal fuzzy random variables is derived as follows
“?” _”A” (“X” )”=” {?(” 0 ,” &”for 0 ? X ? (” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” @(“x” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” )/(“(” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “)” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” ) ” ,” &”for (” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “) ? X ? (” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “)” @?( @?(?(1,@ @(“(” “?” _”1″ “+” ?” ?” ?_”1″ “)(” “?” _”2″ “+” ?” ?” ?_”2″ “)” -“X” )/(“(” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” -“(” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “)” ))@”0,” ))”,” &?(?(“for (” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “) ? X ? (” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “)” @” ” )@?(“for (” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “) ? X ? (” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” @”for X ? (” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” )))?
First, we derive the lower “? -” cut of symmetrical trapezoidal fuzzy random variable is given below:
(?”(X?Y)” ?_”?” ^”L” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “)” )/(“(” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “)” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “)” ) ” ? ?”
(?”(X?Y)” ?_”?” ^”L” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “)” )/(“(” (“?” _”1″ “?” _”2″ )/”2″ “+ ” (“?” _”2″ “?” _”1″ )/”2″ -(?”3?” ?_”1″ “?” _”2″ )/”4″ “)” ) ” ? ?”
?”(X?Y)” ?_”?” ^”L” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “) ? ? (” (“?” _”1″ “?” _”2″ )/”2″ “+ ” (“?” _”2″ “?” _”1″ )/”2″ -(?”3?” ?_”1″ “?” _”2″ )/”4″ “)”
?”(X?Y)” ?_”?” ^”L” -“?” _”1″ “?” _”2″ ” + ” “?” _”1″ “?” _”2″ ” + ” “?” _Theorem: 4.1
If X and Y be two symmetric trapezoidal fuzzy random variables with means “?” _”1″ “,” “?” _”2″ and standard deviations “?” _”1″ “,” “?” _”2″ respectively, then the membership function of multiplication of two symmetric trapezoidal fuzzy random variables is
?”P{((X?Y)” ?_”?” ^”L” – “?” _”1″ “?” _”2″ (“?” /”2″ –”1″ )–”?” _”2″ “?” _”1″ (“?” /”2″ –”1″ )” – ” “?” _”1″ “?” _”2″ (“1 – ” “3?” /”4″ ) –”?” _”1″ “?” _”2″ “) ? 0”
“? ” ?”((X?Y)” ?_”?” ^”U” ” + ” “?” _”1″ “?” _”2″ (“?” /”2″ –”1″ )” + ” “?” _”2″ “?” _”1″ (“?” /”2″ –”1″ )” + ” “?” _”1″ “?” _”2″ (“3?” /”4″ –”1″ )” – ” “?” _”1″ “?” _”2″ “) ? 0}”

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Proof:
The membership function of symmetric trapezoidal fuzzy random variables X and Y with means “?” _”1″ “,” “?” _”2″ and standard deviations “?” _”1″ “,” “?” _”2″ respectively is
?”X = P{(X” ?_”?” ^”L” ” + (1″-“?” /”2″ “)” “?” _”1″ -“?” _”1″ “) ? 0 ? (” “X” _”?” ^”U” -“(1”-“?” /”2″ “)” “?” _”1″ -“?” _”1″ “) ? 0}”
?”Y = P{(Y” ?_”?” ^”L” ” + (1″-“?” /”2″ “)” “?” _”2″ -“?” _”2″ “) ? 0 ? (” “Y” _”?” ^”U” -“(1”-“?” /”2″ “)” “?” _”2″ -“?” _”2″ ” ? 0},”
According to Vahidi. J, Rezvani. S 7, the coordinate points of multiplication of two symmetrical trapezoidal fuzzy random variables is
(“(” “?” _”1″ –”?” _”1″ “)(” “?” _”2″ – “?” _”2″ “),(” “?” _”1″ – “?” _”1″ /”2″ “)(” “?” _”2″ – “?” _”2″ /”2″ “),(” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “),(” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ –”?” _”2″ “),” )
then the membership function of multiplication of two symmetric trapezoidal fuzzy random variables is derived as follows
“?” _”A” (“X” )”=” {?(” 0 ,” &”for 0 ? X ? (” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” @(“x” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” )/(“(” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “)” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” ) ” ,” &”for (” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “) ? X ? (” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “)” @?( @?(?(1,@ @(“(” “?” _”1″ “+” ?” ?” ?_”1″ “)(” “?” _”2″ “+” ?” ?” ?_”2″ “)” -“X” )/(“(” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” -“(” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “)” ))@”0,” ))”,” &?(?(“for (” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “) ? X ? (” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “)” @” ” )@?(“for (” “?” _”1″ ” + ” “?” _”1″ /”2″ “)(” “?” _”2″ ” + ” “?” _”2″ /”2″ “) ? X ? (” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” @”for X ? (” “?” _”1″ ” + ” “?” _”1″ “)(” “?” _”2″ ” + ” “?” _”2″ “)” )))?
First, we derive the lower “? -” cut of symmetrical trapezoidal fuzzy random variable is given below:
(?”(X?Y)” ?_”?” ^”L” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “)” )/(“(” “?” _”1″ -“?” _”1″ /”2″ “)(” “?” _”2″ -“?” _”2″ /”2″ “)” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “)” ) ” ? ?”
(?”(X?Y)” ?_”?” ^”L” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “)” )/(“(” (“?” _”1″ “?” _”2″ )/”2″ “+ ” (“?” _”2″ “?” _”1″ )/”2″ -(?”3?” ?_”1″ “?” _”2″ )/”4″ “)” ) ” ? ?”
?”(X?Y)” ?_”?” ^”L” -“(” “?” _”1″ -“?” _”1″ “)(” “?” _”2″ -“?” _”2″ “) ? ? (” (“?” _”1″ “?” _”2″ )/”2″ “+ ” (“?” _”2″ “?” _”1″ )/”2″ -(?”3?” ?_”1″ “?” _”2″ )/”4″ “)”
?”(X?Y)” ?_”?” ^”L” -“?” _”1″ “?” _”2″ ” + ” “?” _”1″ “?” _”2″ ” + ” “?” _”2″ “?” _”1″ -“?” _”1″ “?” _”2″ -“?(” (“?” _”1″ “?” _”2″ )/”2″ “+ ” (“?” _”2″ “?” _”1″ )/”2″ -(?”3?” ?_”1″ “?” _”2″ )/”4″ “) ? 0″
?”(X?Y)” ?_”?” ^”L” -“?” _”1″ “?” _”2″ (“?” /”2″ -“1” )-“?” _”2″ “?” _”1″ (“?” /”2″ -“1” )-“?” _”1″ “?” _”2″ (“1” -“3?” /”4″ )-“?” _”1″ “?” _”2″ “) ? 0 ”
Which is the required increasing function or lower “? ” – cut of the multiplication of symmetrical trapezoidal fuzzy random variable.
“2” “?” _”1″ -“?” _”1″ “?” _”2″ -“?(” (“?” _”1″ “?” _”2″ )/”2″ “+ ” (“?” _”2″ “?” _”1″ )/”2″ -(?”3?” ?_”1″ “?” _”2″ )/”4″ “) ? 0″
?”(X?Y)” ?_”?” ^”L” -“?” _”1″ “?” _”2″ (“?” /”2″ -“1” )-“?” _”2″ “?” _”1″ (“?” /”2″ -“1” )-“?” _”1″ “?” _”2″ (“1” -“3?” /”4″ )-“?” _”1″ “?” _”2″ “) ? 0 ”
Which is the required increasing function or lower “? ” – cut of the multiplication of symmetrical trapezoidal fuzzy random variable.

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