Download Chapter 2 - Portal UniMAP

Probability and Statistics EQT 272
Semester 2
2015/2016
Discrete Random Variables
• may take on only a countable number of distinct value.
i) The probability of each value of the discrete random variables is between 0 and 1
0  P  X  x  1
ii) The sum of all the probabilities is 1.
 P  X  x  1
xS
iii)
The cumulative distribution function,
F  X   P  X  x
------------------------------------------------------------------------------------------------------------Continuous Random Variables
• takes an infinite number of possible values.
b
P(a  X  b)   f ( x) dx
a
i.
f  x  is nonnegative.
for any real constant a and b with a  b
f x   0
ii. the total area under the curve is equal to 1.

 f x dx 1 .

-------------------------------------------------------------------------------------------------------------
Exercise
1. A random variable Y has the following distribution:
-1
0
1
Y
3C
2C
0.4
PY 
2
0.1
i) The value of the constant C is _________.
ii) Find P  X  0
2. If F 1  0.2, F  2  0.9 , and F  3  1 for a discrete random variable, construct a
probability distribution table for x.
Probability and Statistics EQT 272
Semester 2
2015/2016
3. Let x be the number of cars that a randomly selected auto mechanic repairs on a given
day. The following table lists the probability distribution of x.
x
2
3
4
5
6
P(x)
0.05
0.22
0.40
0.23
0.10
i) What is the probability if the number of cars is 4?
ii) What is the probability if the number of cars is less than 5?
iii) What is the probability if the number of cars at most 3?
iv) What is the probability if the number of cars is at least 4 and less than 6?
v) Determine the cumulative discrete probability distributions.
vi) What is the expected value for the number of cars?
vii) Compute the variance and standard deviation of the number of cars?
4. Given the folllowing probability distribution:
X
0
P  X  x
1
2
Find E  X  and Var  X  .
1
2
3
1
3
1
6
0
5.
Let the probability density function of a random variable Y be
1
-1  y  0
5 ,

1
f  y     cy,
0  y 1
5
otherwise
0,


(i) Find c.
(ii) Find P0  Y  0.5
6.
If X has the probability density function,
 px , 3  x  0
f  x  
.
 px , 0  x  3
i) Find the value of p.
ii) Find P 1  x  2 and P  1  x  1
iii) Find F  x 
iv) Find E  X  and Var  X 