Download STT 231

STT 231 – 01
PRACTICE SHEET SET 5
CONTINUOUS RANDOM VARIABLES AND PROBABILITY DENSITY
FUNCTION, p.d.f.
Example 1: Suppose that the error in the reaction temperature, in oC, for a controlled
laboratory experiment is a continuous random variable X having the function
 x2
 , 1  x  2
f ( x)   3
0,
elsewhere.

(a) Show that f(x) is a density function.
(b) Find P(0 < X < 1).
Example 2
The proportion of people who respond to a certain mail-order solicitation is a continuous
random variable X that has the density function
 2( x  2)
, 0  x 1

f ( x)   5
0,
elsewhere.
(a) Show that P( 0 < X < 1) = 1
(b) Find the probability that more than
1
1
but fewer than
of the people contacted will
4
2
respond to this type of solicitation.
SOME CONTINUOUS DISTRIBUTION FUNCTIONS
1. UNIFORM DISTRIBUTION FUNCTION
A continuous random variable, r.v. X is said to have a uniform distribution on the interval
[a,b] if the probability density function, p.d.f. of X is
 1
, a xb

f ( x)   b  a
0,
otherwise.
1
Example 3: A continuous random variable X that can assume values between x = 1 and x
1
= 3 has a density function given by f ( x)  . (a) Show that the area under the curve is
2
equal to 1. (b) Find P( 2 < X < 2.5). (c) Find P ( X  1.6).
2. EXPONENTIAL DISTRIBUTION FUNCTION
The continuous random variable X has an exponential distribution, with parameter  , if
its density function is given by
for   0.
e x ,
f ( x)  
0,
for x  0
otherwise.
Example 4:
Suppose X is an exponentially distributed random variable with parameter   3 . Find
the following probabilities:
(a) P( X  2)
(b) P( X  1.5)
(c) P( X  3)
(d ) P( X  0.45)
Example 5:
Suppose X has an exponential distribution with   2.5. Find the following probabilities:
(a) P( X  3)
(b) P( X  4)
(c) P( X  1.6)
(d ) P( X  0.4)
3. NORMAL DISTRIBUTION FUNCTION, N ( ,  2 )
Let       and   0 be constants. The density function f given by
  ( x  2)
 e 2
,
f ( x)  

2

0,
2
is called the normal density with parameters 
for    x  
otherwise.
and  .
A special case of the normal density function when   0 and   1 is the standard
normal density function denoted by N(0,1).
2
If X is a continuous random variable, its standard normal density function is given by
 x
 e 2
,   x  
f ( x)  
2

0,
otherwise.
2
DEFINITION: If X is a random variable whose density is normal with parameters
 and  , then,
X 
Z

is a random variable with a standard normal density.
REMARKS: Probably the most important continuous distribution is the normal
distribution which is characterized by its “bell-shaped” curve. The mean is the middle
value of this symmetrical distribution.
When we are finding probabilities for the normal distribution, it is a good idea first to
sketch a bell-shaped curve. Next, we shade in the region for which we are finding the
area, i.e., the probability. [Areas and probabilities are equal] Then use a standard normal
table to read the probabilities.
Example 6:
Let Z have a standard normal distribution N(0,1). Find the following probabilities:
(a) P(0  Z  1.43)
(b) P(0  Z )
(c) P( Z  1.61)
(d ) P(1.52  Z  1.43)
Sketch a bell-shaped curve and shade the area under the curve that equals the
probabilities.
Example 7: DO EXERCISES 20.1, 20.3 TEXT PAGE 138.
REMARK: In statistical applications, we are often interested in right-tail probabilities.
We let z be a number such that the probability to the right of z is  . That is,
P ( Z  z )  
DIAGRAM
3
(b) P(Z  1.64)
Example 8: Find (a) z 0.025
(c) z 0.05
Example 9: Exercises 20.2; Text page 138
MORE PRACTICE PROBLEMS ON DISCRETE AND CONTINUOUS
DISTRIBUTION FUNCTIONS
1. Determine the value c so that each of the following functions can serve as a probability
distribution of the discrete random variable X:
(a)
f ( x)  c( x 2  4)
for x  0,1, 2, 3; [1/30]
(b)
 2  3 

f ( x)  c 
 x  3  x 
for x  0,1, 2. [
1
]
10
2. The shelf life, in days, for bottles of a certain prescribed medicine is a random variable
having the density function
 20,000
, x0

f ( x)   ( x  100) 3
0,
elsewhere.

Find the probability that a bottle of this medicine will have a shelf life of
1
(a) at least 200 days; [ ]
9
(b) anywhere from 180 to 120 days. [0.1020]
3. The total number of hours, measured in units of 100 hours, that a family runs a vacuum
cleaner over a period of one year is a continuous random variable X that has the density
function
0  x 1
 x,

f ( x)  2  x, 1  x  2
0,
elsewhere.

Find the probability that over a period of one year, a family runs their vacuum cleaner
(a) less than 120 hours; [0.68]
4
(b) between 50 and 100 hours. [0.375]
4. A continuous random variable X that can assume values between x = 2 and x = 5 has a
density function given by
2(1  x)
f ( x) 
27
Find
(a) P( X  4)
[
16
]
27
1
[ ]
3
(b) P(3  X  4)
5. Consider the density function
k x
f ( x)  
0,
Evaluate k.
[
0  x 1
elsewhere.
3
]
2
6. Let X denote the amount of time for which a book on two-hour reserve at a college
library is checked out by a randomly selected student, and suppose that X has density
function
x
 ,
f ( x)   2
0,
Calculate
(a) P( X  1)
[[0.25]
0 x2
otherwise.
(b) P(0.5  X  1.5) [0.50]
(c) P(1.5  X ) [0.4375]
7. Suppose the reaction temperature X (in oC) in a certain chemical process has a uniform
distribution with A = - 5 and B = 5.
Compute:
(a) P( X  0)
(b) P(2  X  2)
(c) P(2  X  3)
(d) For k satisfying  5  k  k  4  5, compute P(k  X  k  4) .
5
8. Suppose the distance X between a point target and a shot aimed at the point in a coinoperated target game is a continuous random variable with p.d.f.
 3(1  x 2 )

f ( x)  
4
0,

1  x  1
otherwise
(a) Sketch the graph of f(x).
Compute:
(b) P( X  0) [0.50]
(d ) P( X  0.25 or
(c) P(0.5  X  0.5) [0.6875]
X  0.25) [0.6328]
9. Let X be a random variable with a standard normal distribution. Find
1
(ii) P(0.53  X  2.03) (iii) P( X  0.73) (iv) P ( X  ).
4
10. Let X be normally distributed with mean 8 and standard deviation 4. Find:
(i) P(0.81  X  1.13)
(i) P(5  X  10). (ii) P (10  X  15), (iii) P ( X  15), (iv) P( X  5) .
11. A fair die is tossed 180 times. Find the probability that the face 6 will appear
(i) between 29 and 32 times inclusive, [0.3094] (ii) between 31 and 35 times inclusive.
[0.3245]. The answers provided above are when the Binomial Model is used. What if you
use the Normal Model?
12. Among 10,000 random digits, find the probability that the digit 3 appears at most
950 times. [0.0475]
13. Suppose the temperature T during June is normally distributed with mean 68o and
standard deviation 6o. Find the probability that the temperature is between 70o and 80o.
[0.3479]
14. Suppose the heights H of 800 students are normally distributed with mean 66 inches
and standard deviation 5 inches. Find the number N of students with heights (i) between
65 and 70 inches, [294] (ii) greater than or equal to 6 feet (72 inches) [92].
15. Let X be a random variable with a standard normal distribution. Determine the value
of t if
(i) P(0  X  t )  0.4236 [t = 1.43]
(ii) P( X  t )  .7967
[t = 0.83]
(iii) P(t  X  2)  0.1000
[t = 1.16]
6