Question

Random variables X and Y and joint PMF as follows: $P_{X,Y}(x, y) = \begin{cases} \frac{1}{6x} & x = 1, 2, \dots, 6; \quad y = 1, 2, \dots, x \\ 0 & \text{otherwise} \end{cases}$ Compute the following. HINT: don't forget that you have formulas for the expected value and variance of a uniform random variable. Also, you may need one or both of the following formulas. $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ (a) E[X] = 3.5 (b) E[Y] = 2.25 (c) Var[X] = 2.917 (d) Var[Y] = 1.91 (e) E[XY] = 9.33 (f) Cov[X, Y] = 1.455 (g) What is the correlation coefficient? $\rho_{XY} = 0.616422$

          Random variables X and Y and joint PMF as follows:
$P_{X,Y}(x, y) = \begin{cases} \frac{1}{6x} & x = 1, 2, \dots, 6; \quad y = 1, 2, \dots, x \\ 0 & \text{otherwise} \end{cases}$
Compute the following.
HINT: don't forget that you have formulas for the expected value and variance of a uniform random variable.
Also, you may need one or both of the following formulas.
$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$
(a) E[X] = 3.5
(b) E[Y] = 2.25
(c) Var[X] = 2.917
(d) Var[Y] = 1.91
(e) E[XY] = 9.33
(f) Cov[X, Y] = 1.455
(g) What is the correlation coefficient?
$\rho_{XY} = 0.616422$

Random variables X and Y and joint PMF as follows:
PX,Y(x, y) = (1)/(6x) x = 1, 2, …, 6; y = 1, 2, …, x
0 otherwise
Compute the following.
HINT: don't forget that you have formulas for the expected value and variance of a uniform random variable.
Also, you may need one or both of the following formulas.
∑k=1^n k = (n(n+1))/(2)
∑k=1^n k^2 = (n(n+1)(2n+1))/(6)
(a) E[X] = 3.5
(b) E[Y] = 2.25
(c) Var[X] = 2.917
(d) Var[Y] = 1.91
(e) E[XY] = 9.33
(f) Cov[X, Y] = 1.455
(g) What is the correlation coefficient?
ρXY = 0.616422

Added by Aaron A.

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