Calculate the Expected value, Variance, density and 2nd moment

[alfred2]

I have to solve this exercise. Here are also my solutions but I don't know if they're correct.

Let Ω ⊆ R ^ 2 a finite set of points in R ^ 2. Notation v_i = (x_i, y_i). Pr [] is a probability measure on either Ω.
Random variables X, Y: Ω-> R project a point on the coordinate. For example Y(v_i) = y_i, X (v_i) = x_i.
a) Let n=4 for v_1=(-1,2), v_2 = (3,2), v_3 = (-1, -2), V_4 = (1, -5)and Pr [v_1] = 1 / 3 Pr [v_2] = 1/6, Pr [v_3] = 3/8 and Pr [V_4] = 1/8
i) Determine the density of X, Y and calculate E [X], E [Y]. Density fX: is a function that tells me the probability of a random variable
fX: R -> [0, 1] <=> x -> fX (x): = Pr [X = x], where X is a random variable
Solution:
X (v_1) = -1
X (v_2) = 3
X (v_3) = -1
X (V_4) = 1
Then all possible values ​​of the random variable X are -1,3 and 1 therefore their densities are:
P (X = -1) = P (v_1) + P (v_3) = 17/24
P (X = 3) = P (v_2) = 1/6
P (X = 1) = P (V_4) = 1/8
The same applies to Y
Y (v_1) = 2
Y (v_2) = 2
Y (v_3) = -2
Y (V_4) = -5
Then all possible values ​​of the random variable Y are 2, -2 and -5 therefore their densities would be:
P (Y = 2) = P (v_1) + P (v_2) = 3/6
P (Y = -2) = P (v_3) = 3/8
P (Y = -5) = P (V_4) = 1/8
Then to calculate E [X], E [Y] I just have to apply the formula: Σx * P (X = x) for all possible x of our random variable

ii) Let p = (2, -2) and D the random variable resulting Euclidean distance between a point p, calculate the density D and E [D ^ 2]. Euclidean distance d (v, w) = sqrt ((v_x-W_X) ^ 2 + (v_y-w_y) ^ 2)
Solution:
We calculate the Euclidean distance d (v, w) = sqrt ((v_x-w_x) ^ 2 + (v_y-w_y) ^ 2)
Substituting:
d (v_1, p) = 5
d (v_2, p) = sqrt (17)
d (v_3, p) = 3
d (V_4, p) = sqrt (10)
Then all possible values ​​of the random variable X is 5, sqrt (17), 3, and sqrt (10). But I don't know the probabilities of this random variable. Or they would be the same as the probabilities of v_i, i = 1,2,3,4? To calculate E [D ^ 2] I found this formula
Σx^2*P (X = x) for all possible x our random variableb) Let Ω ⊆ R ^ 2 a finite set, Pr [] a probability measure on Ω any p ^ 2 Dielectric figure any point. D (random variable) gives again the Euclidean distance between any point and p.

i) For which p is E [D ^ 2] Minimum?
Solution:
I'm still lost hereii) How can we represent E [D ^ 2] for this p through the variance.
Solution:
I found this formula and do not know if that's what you need to do: Var [X] = E [X ^ 2] - (E [X]) ^ 2 then E [X ^ 2] = Var [X] + (E [X]) ^ 2.

Someone can say me if I'm on the right track? Thank you =)

 

[Aaron Bex]

Yes, you are on the right track. To answer part b), you need to use the formula for calculating the Euclidean distance between two points and substitute for the values of the random variable D. Then you need to calculate the expected value of D^2 and find the point p for which it is a minimum.
For part ii), you need to use the formula Var[X] = E[X^2] - (E[X])^2 to calculate the variance of the random variable D.