Download A certain genetic characteristic occurs in mice with probability 0

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1. 2/3=0.6667 2.
0
3. 7k/8=7/120.5833
4. 2k/3=4/9=0.4444
6. 0.0803
1
8. 0.75
9.
7.
0.95
5. 0.0803
10. 0
Quiz 3 (2/20/04 in class)
Let X be the next value produced by a random number generator. It is modeled by the following
probability density function (pdf),
0  x 1
k (2  x),
otherwise
0,
f(x)= 
Answer the following 9 questions using this information.
1. Calculate the numeric value of k for this pdf to be legitimate?
1

x2 
1  3k


  k  2   
k
(
2

x
)
dx

k
2
x

1
0

2 0
2 2


1
2.
Calculate P(X>2)?
It is zero without integrating because the largest x is less than 1.
3.
Calculate P(X0.5)?
then k=2/3
0.5

x2 
 1  7k 7

  k 1   
k
(
2

x
)
dx

k
2
x


0

2 0
 8  8 12

0.5
4.
Calculate the expected next value by a random number generator, E(X).
1
 2 x3 
 1  2k 4
  k 1   
E(X)=  xk(2  x)dx  k  x 

3
3
3
9




0
0
1
5.
Calculate the variance of the next value by a random number generator, Var(X).
1
 2x 3 x 4 
 2 1  5k 5
2
E(X )=  x k (2  x)dx  k 
   k    

4 0
 3 4  12 18
 3
0
1
2
Var(X)=5k/12-4k2/9=5/18-16/81=0.0803
6. If Y=X-5, calculate Var(Y).
Same as Var(X)= 5k/12-4k2/9=0.0803
7. Calculate the probability of the next value being within 2 standard deviation of its mean value.
P(-2≤X≤+2)=P(-0.1221 ≤ X ≤ 1.011)=1
8.
What is the probability of the next value being within 2 standard deviation of its mean value using the
chebyshev’s inequality?
0.75
9. What is the probability of the next value being within 2 standard deviation of its mean value using the
empirical rule?
0.95
10. What is the median for standard normal distribution?
Symmetric distribution where the mean is the same as the median