two dices are tossed once. let the random variable be t he sum of the up faces on the dice. A). find and graph the probability distribution of the random variable. and b) calculate the mean (or expectation) of this distribution
X=2=1+1 (1st face + 2nd face)P=1/36
X=3=1+2=2+1 P=2/36
X=4=1+3=3+1=2+2 P=3/36
X=5=1+4=4+1=2+3=3+2 P=4/36
X=6=1+5=5+1=2+4=4+2=3+3 P=5/36
X=7=1+6=6+1=2+5=5+2=3+4=4+3 P=6/36
X=8=2+6=6+2=3+5=5+3=4+4 P=5/36
X=9=3+6=6+3=4+5=5+4 P=4/36
X=10=4+6=6+4=5+5 P=3/36
X=11=5+6=6+5 P=2/36
X=12=6+6 P=1/36
Note 1/36=(1/6)*(1/6),
1/36+2/36+3/36+4/36+5/36+6/36+
5/36+4/36+3/36+2/36+1/36=36/36=1
B) m(X)=2*(1/36)+3*(2/36)+4*(3/36)+
5*(4/36)+6*(5/36)+7*(6/36)+8*(5/36)+
9*(4/36)+10*(3/36)+11*(2/36)+12*(1/36)=
(2+6+12+20+30+42+40+36+30+22+12)/36=
252/36=7
We can also use the independence of numbers on two dice. The mean of the sum
= the sum of two means: if X=X1+X2 and X1, X2 are independent then m(X1+X2)=m(X1)+m(X2)
7=3.5+3.5
To find the probability distribution of the random variable, which represents the sum of the up faces on two dice, we need to determine the probability of each possible outcome. Let's break this down into two parts:
A) Finding the Probability Distribution:
1. List all the possible outcomes when two dice are tossed: The numbers on each die can range from 1 to 6, so there are 6 x 6 = 36 possible outcomes.
2. Calculate the sum of each possible outcome: Write down all the possible sums from 2 to 12. For example, 2 = (1,1), 3 = (1,2), 4 = (1,3), and so on.
3. Determine the probability of each sum: Count the number of times each sum occurs in the list of outcomes. For example, the sum 7 occurs in 6 outcomes (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), while the sum 2 only occurs in 1 outcome (1+1).
4. Calculate the probability for each sum: Divide the number of outcomes for each sum by the total number of possible outcomes (36).
B) Calculating the Mean/Expectation:
The mean (or expectation) of a probability distribution can be found by multiplying each possible value by its corresponding probability and summing them up. In this case, we calculate it as follows:
1. Multiply each sum by their respective probabilities.
2. Sum up all the products obtained.
3. This sum will be the mean of the distribution.
Once you have the probability distribution and mean, you can graph the distribution by plotting each sum on the x-axis and the probability on the y-axis, and then calculate the mean using the steps mentioned above.
Let me know if you need any further assistance in calculating or graphing this distribution!