Exercise 1: Suppose you know that income changes for year 2020 for married couples follow a bi-variate normal distribution. Let the income of men be X and for women Y. Suppose that the population...

Extracted text: Exercise 1: Suppose you know that income changes for year 2020 for married couples follow a bi-variate normal distribution. Let the income of men be X and for women Y. Suppose that the population distribution is (X,Y) N(u, E), where u = (5, 4)' and E = 16 25 9 Assume you randomly pick a random couple: two workers, one male and one female, denote their income gains for 2021 X, and Y;. You know then that their income gains will jointly follow a normal distribution as above. Find the following: 1. the marginal distribution of X; and Y;. 2. P(X; < 10),="" p(y,=""> 3) 3. P(Y, <2| x,="x)," e(y;="" |="" x,="x)" and="" e="" (y2="" |="" x,="x)." 4.="" how="" is="" the="" income="" changes="" for="" married="" women="" distributed="" when="" the="" income="" change="" of="" their="" spouse="" is="" equal="" to="" r.="" for="" which="" values="" of="" r="" is="" the="" distribution="" more="" spread="" out?="" 5.="" find="" p(x;="">< y;)="" and="" p(x,="" +="" y="">< 0)="" i.e.="" 1.="" the="" probability="" that="" the="" female="" worker="" gained="" more="" than="" her="" spouse="" and="" 2.="" the="" probability="" that="" the="" household="" income="" decreased="" in="" 2021.="" 6.="" assume="" you="" have="" a="" random="" sample="" of="" male="" workers="" with="" size="" n,="" what="" is="" distribution="" of="" the="" sample="" mean="" x="E" .="" what="" is="" the="" probability="" that="" the="" sample="" mean="" is="" positive="" (this="" is="" the="" probability="" that="" on="" average="">


Extracted text: income in 2020 increased for men)? You can leave the sample size as n or you can assume that the sample size is n = 25.