Question

Hi, please help me with this exercise, please explain me step by step and please write with very very good calligraphy. Thank you very much.

* From an urn containing 4 white balls and 3 black balls 3 balls are removed with replacement. Following this, a coin is thrown as many times as black balls have been removed and the amount of heads obtained is evaluated. We define X as the random variable that represents the quantity of black balls that are drawn (number of launches of the coin) and Y as the random variable that represents the number of heads obtained in those launches.

a) (6 points) Find the joint probability function g_xy (x,y)
b) (2 points) What is the probability of obtaining the same number of black balls as of heads in the throws?
c) (3 points) Find the marginal probability functions for each of the random variables.
d) (2 points) Determine if the random variables are independent or not.
e) (3 points) If 3 black balls were obtained, what is the probability that at least two heads are obtained?
f) (9 points) Calculate the covariance and the correlation coefficient between X and Y.

* Let X be the random variable representing the temperature, measured in degrees Celsius, at which a certain reaction begins to happen, and let Y be the random variable representing the reaction time, mediated in seconds. It is known that its joint probability function is given by:

fw(x,y) = In other case.

a) (5 points) Graph the range of the joint probability density function. Calculate the value of the constant k so that the joint probability density function of the random variables X and Y is correctly defined.
b) (4 points) Find the marginal probability density functions for the random variables X and the variable Y.
c) (3 points) If the reaction time is 1.5 seconds, what is the expected value of the reaction temperature?
d) (3 points) If it is known that the reaction time is 1.5 seconds, what is the probability that the reaction temperature is between 0 oC and 2 oC?
e) (4 points) Find the covariance of the random variables X and Y.
f) (3 points) Are temperature and reaction time independent random variables?

Thank you very much!

Answer 1

solving first 4

4 white and 3 black balls

we choose 3 balls

so

probabilty of 0 black 3 white p(x=0) = 3^0*4^3/7^3 =64/343

probabilty of 1 black 2 white p(x=1) = 3c2*3^1*4^2/7^3 =144/343

2 black 1white 3c2*3^2*4/7^3 p(x=2) = 108/343

3 black 0 white 3c# 3^3/7^3 p(x=3) = 27/343

we can have 0, 1,2,3 head

0 heads in 1 toss 1/2

1 heads in 1 toss 1/2

0 heads in 2 toss 2C0 *(1/2)^0*(1/2)^2 =1/4

1 heads in 2 toss 2C1 *(1/2)*(1/2) = 1/2

2 heads in 2 toss 2C2 *(1/2)^2 (1/2)^0= 1/4

0 heads in 3toss 3C0 *(1/2)^0(1/2)^3 = 1/8

1 heads in 3 toss 3C1 *(1/2)^1(1/2)^2 = 3/8

2 heads in 3 toss 3C2 *(1/2)^2(1/2)^1 = 3/8

3 heads in 3 toss 3C3 *(1/2)^3(1/2)^0= 1/8

we multply each possibility of heads with the number of black balls to get

putting this in a table

heads/black	0	1	2	3	total
0	0.186588921	0.209912536	0.078717201	0.00983965	0.485058309
1	0	0.209912536	0.157434402	0.02951895	0.396865889
2	0	0	0.078717201	0.02951895	0.108236152
3	0	0	0	0.00983965	0.00983965
total	0.186588921	0.419825073	0.314868805	0.078717201	1

sum of diagonal elements of table.485 probabilty of same number of heads as black

heads		black
0	0.485058309	0	0.186588921
1	0.396865889	1	0.419825073
2	0.108236152	2	0.314868805
3	0.00983965	3	0.078717201

c)

marginal probabiltiy total of rows and coumns

P(x,y) = p(x)p(y) for independenced)

so lets check for p(x=1, y=1) =p(x=1) p(y=1)

p(x=1, y=1) =.2099

p(x=1)p(y=1) =.3968*.4198 . =0.1665766

as both are unequal the varables are not independent

e) using bayes theorem

P(y>=2|x=3)= p(y=2|x=3) +p(y=3|x=3) = (P(y=2, x=3)+p(y=3, x=3)/ p(x=3)

=(.0295+.00985)/.0787 = .5