1. If one bronze coin is thrown and the front side appears, it is expressed as 1, and if the back side appears, it is expressed as 0. Answer the following questions, setting the random variables for the appearance of each bronze coin table as when throwing three bronze coins.
2. The probability density of is given by the
3. Let the random variables be the rolls of the two dice
1.
a Let . Then the domain of is .The probabilty istribution is given by . Consider . ,It's the same as throwing three bronze coins and finding the probability that all tails come out. All tail combinations are 1 out of 8.Thus, .Similarly for ,
2.
a To find the probability density function of , examine the distribution function of and use the relation of .
The distribution function of is , and the distribution function of is given by
The expectation of is . Then to find , it is enough to find . . Then
Note : Since the probability density function of was calculated, directly calculate by .
Finally, we find the variance of .Note that . Then to find , we need to find .
b To find the probability density function of , examine the distribution function of and use the relation of
The distribution function of is , and the distribution function of is . Since it is given by,
Thus the probability density function of is
We next find the expectation..
.
Finally, find the variance of .
3.
a Let . Then are independent and the expectation is given by