X is a random variable with mean X 401 and standard deviati

X

is a random variable with mean

_X

= 40.1 and standard deviation

_X

= 6.1, while

Y

is a random variable (independent of

X

) with mean

_Y

= 55 and standard deviation

_Y

= 8.4.

a) Let

Z = X+Y

. That is,

Z

is the random variable representing the sum of

X

and

Y

. Find the mean and standard deviation of

Z

. Give your answers to two decimal places.

_Z

=

_Z

=
b) Let

W = X+X

. That is,

W

is the random variable representing the sum of two independent outcomes of

X

. Find the mean and standard deviation of

W

. Give your answers to two decimal places.

_W

=  

_W

=
c) Let

R = 2X

. That is,

R

is the random variable representing 2 times a single outcome of

X

. Find the mean and standard deviation of

R

. Give your answers to two decimal places.

_R

=  

_R

=
d) Let

T = 2X+3Y

. Find the mean and standard deviation of

T

. Give your answers to two decimal places.

_T

=  

_T

=
e) Let

S = 0.2X+0.8Y

. Find the mean and standard deviation of

S

. Give your answers to two decimal places.

_S

=  

_S

=

Solution

X is a random variable with mean _X = 40.1 and standard deviation _X = 6.1, while

Y is a random variable (independent of X) with mean _Y = 55 and standard deviation _Y = 8.4.

a) Let

Z = X+Y

. That is, Z is the random variable representing the sum of X and Y

. Find the mean and standard deviation of Z

. Give your answers to two decimal places.

E(X+Y) = E(X)+E(Y)

VAR(X+Y)=VAR(X)+VAR(Y)

VAR(X)=6.1² = 37.21

VAR(Y)= 8.4²=70.56

_Z = 40.1+55=95.1

    _Z = SQRT(37.21+70.56) =10.38

b) Let

W = X+X

. That is, W is the random variable representing the sum of two independent outcomes of X

. Find the mean and standard deviation of W

. Give your answers to two decimal places.

_W =   40.1+40.1 =80.2

_W = SQRT(37.21+37.21) =8.63

c) Let

R = 2X

. That is, R is the random variable representing 2 times a single outcome of X

. Find the mean and standard deviation of R

. Give your answers to two decimal places.

E(aX) = aE(X)

VAR(aX) =a²VAR(X)

Sd(aX) = aSd(X)

_R =   2*40.1 =80.2

_R   2*6.1=12.2

d) Let T = 2X+3Y

. Find the mean and standard deviation of T. Give your answers to two decimal places.

_T =  2*40.1+3*55 =245.2

_T =    SQRT( 2²*37.21+3²*70.56) = 28.00

e) Let

S = 0.2X+0.8Y

. Find the mean and standard deviation of S

. Give your answers to two decimal places.

_T =  0.2*40.1+0.8*55 =52.02

_T =    SQRT( 0.2²*37.21+0.8²*70.56) = 6.83