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
(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
(a) Show that P( 0 < X < 1) = 1
(b) Find the probability that more than but fewer than of the people contacted will 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
Example 3: A continuous random variable X that can assume values between x = 1 and x = 3 has a density function given by (a) Show that the area under the curve is equal to 1. (b) Find P( 2 < X < 2.5). (c) Find
2. EXPONENTIAL DISTRIBUTION FUNCTION
The continuous random variable X has an exponential distribution, with parameter , if its density function is given by
for
Example 4:
Suppose X is an exponentially distributed random variable with parameter . Find the following probabilities:
Example 5:
Suppose X has an exponential distribution with Find the following probabilities:
3. NORMAL DISTRIBUTION FUNCTION,
Let and be constants. The density function f given by
is called the normal density with parameters
A special case of the normal density function when is the standard normal density function denoted by N(0,1).
If X is a continuous random variable, its standard normal density function is given by
DEFINITION: If X is a random variable whose density is normal with parameters then,
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:
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 be a number such that the probability to the right of is . That is,
DIAGRAM
Example 8: Find
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:
[1/30]
[]
2. The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function
Find the probability that a bottle of this medicine will have a shelf life of
(a) at least 200 days;
(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
Find the probability that over a period of one year, a family runs their vacuum cleaner
(a) less than 120 hours; [0.68]
(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
Find
5. Consider the density function
Evaluate k. []
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
Calculate
7. Suppose the reaction temperature X (in oC) in a certain chemical process has a uniform distribution with A = - 5 and B = 5.
Compute:
(d) For k satisfying compute .
8. Suppose the distance X between a point target and a shot aimed at the point in a coin-operated target game is a continuous random variable with p.d.f.
(a) Sketch the graph of f(x).
Compute:
9. Let X be a random variable with a standard normal distribution. Find
(i) (ii) (iii) (iv)
10. Let X be normally distributed with mean 8 and standard deviation 4. Find:
(i) (ii) (iii) (iv) .
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) [t = 1.43]
(ii) [t = 0.83]
(iii) [t = 1.16]
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