# Find the Variance

is known that the Tomato crop (in ton) in some farm are Sampled for 10 years.
the Standard deviation of the crop was 2 ton.
the Income (Y) from the Tomato Depends on the crop (X)
according to following connection Y=3X-2 the Variance Income from the Tomato in this Sampled is 4?

if i know that the Standard deviation is 2 so i just need to do 2^2 to get the Variance yes?

and 1 more Question: the Value of Pearson product-moment correlation coefficient between the crop and the Income is?
how i can solve it?

thanks for help.

vela
Staff Emeritus
Homework Helper
is known that the Tomato crop (in ton) in some farm are Sampled for 10 years.
the Standard deviation of the crop was 2 ton.
the Income (Y) from the Tomato Depends on the crop (X)
according to following connection Y=3X-2 the Variance Income from the Tomato in this Sampled is 4?

if i know that the Standard deviation is 2 so i just need to do 2^2 to get the Variance yes?
If you're referring to var(Y), then no, it isn't equal to 4.
and 1 more Question: the Value of Pearson product-moment correlation coefficient between the crop and the Income is?
how i can solve it?

thanks for help.

ok so how can i solve it?
and is told that the Standard deviation are 2
so for example if i know the Variance and it 4 so to find the Standard deviation i just need to do sqrt4.
and in this case i know the Standard deviation so to find the Variance i need to do 2^2 instead of sqrt4. no?

thank you.

ok so how can i solve it?
and is told that the Standard deviation are 2
so for example if i know the Variance and it 4 so to find the Standard deviation i just need to do sqrt4.
and in this case i know the Standard deviation so to find the Variance i need to do 2^2 instead of sqrt4. no?

thank you.

vela
Staff Emeritus
Homework Helper
You're given stdev(X)=2 tons. You can't just square it to find the variance of Y, which is a different random variable.

so can you give me any way to solve it or give me direction ?

thanks

vela
Staff Emeritus
Homework Helper
Look up in your textbook how the variances of X and Y are related.

well is will be correct to say:
if i use the formula var(aX+b)=a^2var(X) and then i will put numbers and get this:
9*4=36 and this is the Variance Income from the Tomato in this Sampled?

thanks.

HallsofIvy
Homework Helper
Yes,that is correct. Are you interested in how that formula is derived?

For the variable x, the mean is given by
$$\mu_x= \frac{\sum x_i}{n}$$
while, for the variable y, it is
$$\mu_y= \frac{\sum y_i}{n}$$.

If y= ax+ b, that is the same as
$$\mu_y= \frac{\sum (ax_i+ b)}{n}= \frac{a\sum x_i+ \sum b}{n}= a\frac{\sum x_i}{n}+ \frac{nb}{n}[/itex]$= a\mu_x+ b$$. For the variable x, the variance is given by $$v_x= \frac{\sum (x_i- \mu_x)^2}{n}$$ while, for the variable y, it is $$v_y= \frac{\sum (y_i- \mu_y)^2}{n}$$. Again, if y= ax+ b, that is $$v_y= \frac{\sum (ax_i+ b- (a\mu_x+ b))^2}{n}=$[itex] \frac{\sum{(ax_i- a\mu_x)^2}{n}$$
$$= \frac{a^2(x_i- \mu_x)^2}{n}= a^2\frac{(x_i- \mu_x)^2}{n}= a^2 v_x$$.

WOW.
thanks for show all this way.