Joint probability

In summary, joint probability is a statistical measure that calculates the likelihood of two or more events occurring simultaneously. It is calculated by multiplying the individual probabilities of each event. It differs from conditional probability, which focuses on the probability of one event occurring given another event has already occurred. Joint probability is commonly used in finance, economics, and social sciences to analyze the relationship between multiple variables. However, it has limitations, such as assuming independence between events and becoming more complex as the number of events increases.
  • #1
Lucy77
10
0
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
 
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  • #2
erm, evidently you just want us to do your homework not explain any maths. there's a homework section for that. my advice? use the formula
 
  • #3
for sharing this joint probability table! To answer your questions:

a) P(Y1=0) = 35/120 = 7/24

b) P(Y2=1 | Y1=1) = P(Y2=1 and Y1=1) / P(Y1=1) = (1/4) / (63/120) = 2/3

c) P(Y2=1) = (56/120) + (56/120) + (8/120) = 120/120 = 1

e) To determine if Y1 and Y2 are independent or dependent, we can compare the individual probabilities to the joint probabilities. If the joint probabilities are equal to the product of the individual probabilities, then the variables are independent. In this case, we see that P(Y1=0) * P(Y2=1) = (7/24) * (56/120) = 1/10, which is not equal to P(Y1=0 and Y2=1) = 1/4. Therefore, Y1 and Y2 are dependent.

f) To find the correlation coefficient r, we can use the formula r = (E(XY) - E(X)E(Y)) / √(Var(X)Var(Y)). From the table, we can calculate:

E(XY) = (0*0)*(1/12) + (0*1)*(1/6) + (0*2)*(1/24) + (1*0)*(1/4) + (1*1)*(1/4) + (1*2)*(1/40) + (2*0)*(1/8) + (2*1)*(1/20) + (2*2)*(21/120) + (3*1)*(1/120) = 111/120

E(X) = (0)*(1/12) + (1)*(1/4) + (2)*(1/8) + (3)*(1/120) = 11/12

E(Y) = (0)*(1/12) + (1)*(1/6) + (2)*(1/20) + (1)*(1/4) + (1)*(1/4) + (2)*(1/40) + (0)*(1/24) + (1)*(1/20) + (2)*(
 

1. What is joint probability?

Joint probability is a statistical measure that calculates the likelihood of two or more events occurring simultaneously. It is often used in the fields of probability theory and statistics to analyze the relationship between multiple variables.

2. How is joint probability calculated?

Joint probability is calculated by multiplying the individual probabilities of each event occurring. For example, if the probability of event A is 0.5 and the probability of event B is 0.3, the joint probability of both events occurring is 0.5 x 0.3 = 0.15.

3. What is the difference between joint probability and conditional probability?

Joint probability considers the likelihood of multiple events occurring simultaneously, while conditional probability focuses on the probability of one event occurring given that another event has already occurred. Joint probability is calculated by multiplying probabilities, while conditional probability is calculated by dividing probabilities.

4. How is joint probability used in real life?

Joint probability is used in a variety of fields, including finance, economics, and social sciences, to analyze the relationship between multiple variables. For example, it can be used to determine the probability of a stock portfolio generating a certain return, or the likelihood of a certain demographic group purchasing a product.

5. What are some limitations of joint probability?

One limitation of joint probability is that it assumes independence between events, meaning that the occurrence of one event does not affect the likelihood of the other event. In real life, events are often interdependent, which can affect the accuracy of joint probability calculations. Additionally, joint probability can become more complex and difficult to calculate as the number of events increases.

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