1. Gohan throws a fair tetrahedral die with faces numbered 1,2,3,4. If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable X denote Gohan’s score.
a. Show that P(X=2)= 5/16
b. The table below shows the probability distribution of X.
|
x |
2 |
3 |
4 |
5 |
6 |
7 |
|
P(X=x) |
5/16 |
1/16 |
3/8 |
1/8 |
1/16 |
1/16 |
Calculate E(X) and Var(X)
2. The probability distribution of the random variable X is shown in the following table
|
x |
-2 |
-1 |
0 |
1 |
2 |
3 |
|
P(X=x) |
0.08 |
p |
0.12 |
0.16 |
q |
0.22 |
The mean of X is 1.05.
a. Write down two equations involving p and q and hence find the values of p and q
b. Find the variance of X
3. The probability distribution of the discrete random variable X is shown in the table below:
|
x |
-3 |
-1 |
0 |
4 |
|
P(X=x) |
a |
b |
0.15 |
0.4 |
Given that E(X) = 0.75, find the values of a and b
4. The discrete random variable X takes the values 1,4,5,7 and 9 only. The probability distribution of X is shown in the table.
|
x |
1 |
4 |
5 |
7 |
9 |
|
P(X=x) |
4p |
5p2 |
1.5p |
2.5p |
1.5p |
Find p.
5. In a probability distribution the random variable X takes the value x with probability kx, where x takes values 1, 2, 3, 4, 5 only.
a. Draw up a probability distribution table for X, in terms of k, and find the value of k
b. Find E(X)
6. A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random.
7. The random variable X represents the number of geese chosen.
a. Draw up the probability distribution of X
b. Show that E(X)= 8/7 and calculate Var(X)
8. The random variable X has the probability distribution shown in the table.
|
x |
2 |
4 |
6 |
|
P(X=x) |
0.5 |
0.4 |
0.1 |
Two independent values of X are chosen at random. The random variable Y takes the value 0 if the two values of X are the same.
Otherwise the value of Y is the larger value of X minus the smaller value of X
a. Draw up the probability distribution table for Y
b. Find the expected value of Y
9. A team of 4 is to be randomly chosen from 3 boys and 5 girls. The random variable X is the number of girls in the team.
a. Draw up a probability distribution table for X
b. Given that E(X) = 5/2, calculate Var(X)
10. Two ordinary fair dice are thrown. The resulting score is found as follows.
· If the two dice show different numbers, the score is the smaller of the two numbers.
· If the two dice show equal numbers, the score is 0
a. Draw up the probability distribution table for the score
b. Calculate the expected score.
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