1. Tutorial 7 Probability and Calculus
MT129 – Calculus and Probability

2. Outline Random Variables
Discrete & Continuous Probability Distributions
Cumulative Distribution Functions
Mean or Expected Value
Variance & Standard Deviation
Applications of Expected Value, Variance & Standard Deviation
MT129 – Calculus and Probability

3. Random Variables
e.g.
Suppose that a coin is flipped three times. Let X(t) be the random variable that equals the number of heads that appear when t is the outcome. Then X(t) takes on the following values:
X(HHH) = 3, X(TTT) = 0,
X(HHT) = X(HTH) = X(THH) = 2,
X(TTH) = X(THT) = X(HTT) = 1.
Let X be the random variable defined by the waiting time, in hours, between successive speeders spotted by a radar unit. The random variable X takes on all values x for which x ≥ 0.
MT129 – Calculus and Probability

4. Discrete & Continuous Sample Spaces
MT129 – Calculus and Probability

5. Discrete & Continuous Probability Functions
MT129 – Calculus and Probability

6. Discrete & Continuous Probability Functions
In the case of tossing a coin three times, the variable X, representing the number of heads, assumes the value 2 with probability 3/8, since 3 of the 8 equally likely sample points result in two heads and one tail. x 1 2 3
P(X = x) = f (x)
1 / 8
3 / 8
Suppose that the error in the reaction temperature, in ◦C, for a controlled laboratory experiment is a continuous random variable X having the probability density function
MT129 – Calculus and Probability

7. Cumulative Distribution Functions
MT129 – Calculus and Probability

8. Example 1
Suppose that the error in the reaction temperature, in ◦C, for a controlled laboratory experiment is a continuous random variable X having the probability density function
Verify that f (x) is a density function.
Find F(x), and use it to evaluate P(0 < X ≤ 1).
Solution:
(a)
(b)
MT129 – Calculus and Probability
3 - 8

9. Example 2
Solution:
3 - 9
MT129 – Calculus and Probability
3 - 9

10. Example 3
Solution:
3 - 10
3 - 10
MT129 – Calculus and Probability
3 - 10

11. Example 4
Solution:
3 - 11
3 - 11
MT129 – Calculus and Probability
3 - 11

12. Mean or Expected Value
MT129 – Calculus and Probability

13. Variance and Standard Deviation
MT129 – Calculus and Probability

14. Example 1
Find expected value, variance, and the standard deviation of X, where X is the random variable whose probability table is given in Table 5.
Solution
MT129 – Calculus and Probability

15. Example 2
The weekly demand for a drinking-water product, in thousands of liters, from a local chain of efficiency stores is a continuous random variable X having the probability density
Find the mean and variance of X.
Solution:
MT129 – Calculus and Probability

16. Applications of Expected Value
Example: The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let X be the outcome. Then X is a random variable taking on the values 0, 1, and 2.
Write out a probability table for X.
Compute E(X).
Interpret E(X).
Solution:
Number of Calls 1 2
Probability
30/60 = 1/2
20/60 = 1/3
10/60 = 1/6
)a)
)b)
)c) Since E(X) = 0.67, this means that the expected number of phone calls in a 1-minute period is 0.67 phone calls.
MT129 – Calculus and Probability