Download A random variable is discrete if it has a finite or a countable number

TUTORIAL 3 ( Random Variable)
A random variable is discrete if it has a finite or a countable number of possible outcomes.
A random variable is continuous if it has an uncountable number of possible outcomes, represented by an
interval on the number line.
A discrete probability distribution must satisfy the following conditions:
 The probabilty of each each values of the discrete random variable is between 0 and 1.
That is, 0  P X  x  1 .


 The sum of all the probabilities is 1. That is,
 P X  x   1 .
xS
f x  0 such that
X is a continuous random variable if there exist a function
Pa  X  b  

 f x dx .

A continuous probability distribution must satisfy the following conditions:
 f x  0 . That is, f is nonnegative.



 f x dx 1 . That is, the total area under its graph is 1.

The mean of a random variable X is also known as the expected value of X and written as
 If X is discrete,
 x  E  X    xP X  x 
   x  EX 
xS
 If X is continuous,
 x  E  X    xf x dx
 Var  X   E X 2   E  X 
2
1) Determine whether the following random variables are discrete or continuous.
a) Number of calls a lawyer makes in one day.
___________
b) The amount of rain that fell in the month of December.
___________
c) Number of rainy days in the month of December.
___________
d) The length of time a dentist spent on his patient in a day. ___________
2) Decide whether each distribution is a probability distribution.
a)
-1
0
2
X
0.28
0.21
0.3
P X  x
4
0.25
b)
Y
PY  y 
2.5
0.28
3
0.21
7.8
-0.3
10.5
0.81
3) A random variable Y has the following distribution:
Y
PY 
-1
3C
0
2C
The value of the constant C is _________.
1
0.4
2
0.1
4) Determine the value of c so that the following function is a probability function for a
discrete random variable.

f x   c x 2  2

for
x  0,1,2
5) A survey asked a sample of people living in Kangar, how many times they shop at The
Store supermarket each week and the following distribution is obtained.
0
X
P X  x 0.20
1
0.25
2
0.25
3
0.10
4
0.09
5
0.06
6
0.03
7
0.02
a) Verify whether this distribution is a probability distribution.
b) Find P X  3
c) Find P2  X  6
The following two questions refer to the following situation.
All human blood can be “ABO” typed as belonging to one of A, B, O, or AB types. The
actual distribution varies slightly among different groups of people, but for a randomly
chosen person from North America, the following are the approximate probabilities:
Blood type O
Probability .45
A
.40
B
.11
AB
.04
6) Consider an accident victim with type B blood. She can only receive a transfusion from a
person with type B or type O blood. What is the probability that a randomly chosen person
will be suitable donor?
7) What is the probability that both people in a couple will have the same blood type if
mating are random with respect to blood type, i.e. one partner’s blood type does not
influence the blood type of the other partner.
8) Suppose a random variable X has the following cumulative distribution function
X
F x 
1
0.11
2
0.25
a) Find P X  3
b) Find P X  4
c) Find the distribution of X.
3
0.64
4
0.77
5
1
9) Let X be a continuous random variable with the following probability density function.


c x 2  1 ,
f x   
0,
0  x 1
otherwise
a) Evaluate c.
b) Find P0  X  0.5
10) Let the probability density function of a random variable Y be
1
5 ,

1
f  y     cy,
5
0,


-1  y  0
0  y 1
otherwise
a) Find c.
b) Find P0  Y  0.5
11) The total amount of gasoline (measured in ten thousands of gallons), pumped at a certain
gas station in a month is a random variable X with the following probability density
function.
0  x 1
 x,

f x   2  x,
1 x  2
0,
otherwise

a) Find F  X 
b) What is the probability that the station pumps between 9000 and 11000 gallons in a
month.
12) Suppose a random variables X has the following distribution
X
P X  x
2
0.3
a) Find the expected value of X.
b) Find the variance of X.
c) Find the standard deviation of X.
4
0.5
8
0.2
13) Find the mean number of times people living in Kangar shop at The Store supermarket.
The data is given in question 5.
14) A restaurant manager is considering a new location for her restaurant. The projected
annual cash flow for the new location is:
Annual Cash Flow
Probability
$10,000
0.10
$30,000
0.15
$70,000
0.50
$90,000
0.15
$100,000
?
The expected cash flow for the new location is:
15) A business evaluates a proposed venture as follows. It stands to make a profit of $10,000
with probability 3/20, to make a profit of $5,000 with probability 9/20, to break even with
probability 1/4 and to lose $5,000 with probability 3/20. The expected profit in dollars is:
16)
Let X be a continuous random variable with the following probability density function
1
 2  x ,
f x    2
0,
a) Find E X 
b) Find Var X 
0  x 1
otherwise