Answer:
a
[tex]E[(3X+1)^2]= 8.5 [/tex]
b
[tex]P(1 < X < 2)=0.1170 [/tex]
Step-by-step explanation:
From the question we are told that
The parameter of X is [tex]\lambda = 2[/tex]
Generally the expected value of X is
[tex]E(X) = \frac{1}{\lambda }[/tex]
[tex]E(X) = \frac{1}{2}[/tex]
=> [tex]E(X) = 0.50 [/tex]
Generally we have that
[tex]E(X^2) = E(X)^2 + E(X)^2[/tex]
=> [tex]E(X^2) = [\frac{1}{2}] ^2 + [\frac{1}{2} ]^2[/tex]
=> [tex]E(X^2) = 0.5 [/tex]
Generally
[tex]E[(3X+1)^2]= E(9x^2 + 1 + 6x)[/tex]
=> [tex]E[(3X+1)^2]= 9E[X^2] + 1 + 6 E[X])[/tex]
=> [tex]E[(3X+1)^2]= 9* 0.5 + 1 + 6 * 0.5 [/tex]
=> [tex]E[(3X+1)^2]= 8.5 [/tex]
Generally
[tex]P(1 < X < 2)= P(X < 2) - P(X < 1)[/tex]
Here [tex]P(X < 2 ) = e^{- 2 * \lambda }[/tex]
=> [tex]P(X < 2 ) = e^{- 2 * 2 }[/tex]
=> [tex]P(X < 2 ) = e^{- 4}[/tex]
and
[tex]P(X < 1 ) = e^{- 1 * \lambda }[/tex]
[tex]P(X < 1 ) = e^{- 1 * 2 }[/tex]
[tex]P(X < 1 ) = e^{-2 }[/tex]
So
[tex]P(1 < X < 2)= e^{-2 } - e^{- 4} [/tex]
[tex]P(1 < X < 2)=0.1170 [/tex]
