Suppose Company 1's stock return 𝑋𝑋 is a random variable and takes three possiblevalues: {-0.1, 0.1, 0.2}. And Company 2's stock return 𝑌𝑌 is a random variable and takes twopossible values: {-0.3, 0.4}. The joint probability distribution 𝑓𝑓(𝑋𝑋, 𝑌𝑌) is given as follows:𝑓𝑓(−0.1, −0.3) = 0.1, 𝑓𝑓(0.1, −0.3) = 0.2, 𝑓𝑓(0.2, −0.3) = 0.2,𝑓𝑓(−0.1,0.4) = 0.2, 𝑓𝑓(0.1,0.4) = 0.2, 𝑓𝑓(0.2,0.4) = 0.1.Please calculate the following:(a) Marginal distributions: 𝑓𝑓𝑋𝑋(𝑥𝑥) and 𝑓𝑓𝑌𝑌(𝑦𝑦). (4 points)(b) Mean: 𝐸𝐸(𝑋𝑋) and 𝐸𝐸(𝑌𝑌). (4 points)(c) Variance: 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋) and 𝑉𝑉𝑎𝑎𝑟𝑟(𝑌𝑌). (4 points)

uppose Company 1’s simple return X is a random variable and takes three possiblevalues {−0.1, 0.1, 0.2}. Company 2’s simple return Y is a random variable and takes twopossible values {−0.3, 0.4}. The joint probability distribution f (X, Y ) is as follows:f (X = −0.1, Y = −0.3) = 0.2, f (X = 0.1, Y = −0.3) = 0.2, f (X = 0.2, Y = −0.3) = 0.1,f (X = −0.1, Y = 0.4) = 0.1, f (X = 0.1, Y = 0.4) = 0.2, f (X = 0.2, Y = 0.4) = 0.2Please calculate the following:(a). Marginal distributions: fX (x) and fY (y);(b). Mean: E[X] and E[Y ];(c). Variance: var(X) and var(Y );(d). Covariance: cov(X, Y );(e). Correlation: corr(X, Y )

Random variable X may take value -1 or 1; Y may take value 1, 2 or 3. The following Table shows the joint probability of random variables X and Y.Y1 2 3X -1 0.1 0.2 0.31 0.2 0.1 0.1Table 2: Joint probability distribution of X and Y. For example, the probability of X=-1 and Y=1 is 0.1. Provide solution to all questions below.(a) Marginal probability distribution of X(b) The expected value, variance and standard deviation of X. (c) Marginal probability distribution of Y(d) The expected value, variance and standard deviation of Y. (e) Conditional probability distribution of Y given X=-1.(f) The conditional expected value, conditional variance and conditional standard deviation of Y

Let the discrete RV 𝑋~𝑈[−2,2] (Uniform dist.). Let 𝑌 = 𝑋2a) What values X and Y can take? Find pdf’s of both X and Y.b) Compute the joint pdf, 𝑓𝑋𝑌(𝑥𝑖, 𝑦𝑖)c) Compute the E(X) and E(Y)d) Compute the Cov(X,Y)e) Compute the 𝜌𝑋𝑌 = 𝐶𝑜𝑟(𝑋, 𝑌).f) Are X and Y independent? Prove it.

eet the joint p.d.f. of X1 and X2 be:ℎ(𝑥1, 𝑥2) = {8𝑥1𝑥2 for 0 < 𝑥1 < 𝑥2 < 10 otherwisea) Find the joint p.d.f. of 𝑌1 = 𝑋1𝑋2and 𝑌2 = 𝑋2b) Are 𝑌1 and 𝑌2 independent? Why? (10 points

You are given the following return probability distribution for Stock X and Y:  Bear market Normal market Bull marketProbability 0.2 0.5 0.3Stock X -10% 10% 20%Stock Y -5% 20% 10%What is the return correlation between Stock X and Y?Group of answer choices0.41930.62250.54470.2071