How Does Joint Probability Density Determine Dependence Between Variables?

[Lucy77]

If the values of the joint probability density of Y1 and Y2 are as shown:

0 1 2 total

0 1/12 1/6 1/24 35/120
1 1/4 1/4 1/40 63/120
2 1/8 1/20 ... 21/120
3 1/120 ... ... 1/20
ttl 56/120 56/120 8/120 1

whew ;-)

ok Find

a) P (Y1=0)
b) P(Y2=1 | Y1=1)
c P(Y2=1)
e Check if Y1 and Y2 are independent or dependent
f Evaluate the correlation coefficient r of y1 and y2

Thanks so much

 

[HallsofIvy]

Unfortunately, it is not clear from your chart whether Y1 is "across" or "down"!

Assuming that Y1 can be 0, 1, 2, or 3 (down) and that Y2 can be 0, 1, or 2, (across) then

P(Y1= 0) is just the sum of the probabilities in the first row (first column if I have Y1, Y2 backwards).

P(Y2=1 | Y1=1) is the probability in the "1 -1" cell divided by P(Y1= 1).

P(Y2= 1) is the sum of the probabilities in the second column (second row if I have Y1, Y2 backwards)

 

[M98Ranger]

for providing the joint probability density table! Let's use this information to answer the questions.

a) P(Y1=0) is the probability that Y1 takes on the value of 0. Looking at the first row of the table, we can see that this probability is 1/24 or 0.0833.

b) P(Y2=1 | Y1=1) is the conditional probability that Y2 takes on the value of 1, given that Y1 is equal to 1. From the table, we can see that the probability of Y2=1 and Y1=1 is 1/4 or 0.25. The probability of Y1=1 is 1/4 or 0.25. Therefore, the conditional probability is 0.25/0.25 = 1.

c) P(Y2=1) is the marginal probability that Y2 takes on the value of 1, regardless of the value of Y1. From the table, we can see that the sum of the second column is 56/120 or 0.4667. This is the probability of Y2=1.

e) To check if Y1 and Y2 are independent, we can compare the joint probabilities to the product of the marginal probabilities. If they are equal, then Y1 and Y2 are independent. Let's look at Y1=0 and Y2=1. The joint probability of these two events is 1/12, while the product of the marginal probabilities is (1/24) x (56/120) = 7/120. Since these two values are not equal, we can conclude that Y1 and Y2 are dependent.

f) The correlation coefficient r of Y1 and Y2 can be calculated using the formula r = cov(Y1,Y2) / (σ1 * σ2). Cov(Y1,Y2) represents the covariance between Y1 and Y2, and σ1 and σ2 represent the standard deviations of Y1 and Y2, respectively. From the table, we can calculate that the covariance is -1/120 and the standard deviations are √(56/120) and √(56/120). Therefore, the correlation coefficient is -1/√(56/120)^2 = -0.25. This indicates a moderate negative correlation between Y1 and Y2.