1. Lecture note 6
Continuous Random Variables and Probability distribution

2. Continuous Random Variables
A random variable is continuous if it can take any value in an interval.

3. Probability Distribution of Continuous Random Variables
For continuous random variable, we use the following two concepts to describe the probability distribution
Probability Density Function
Cumulative Distribution Function

4. Probability Density Function
Probability Density Function is a similar concept as the probability distribution function for a discrete random variable.
You can consider the probability density function as a “smoothed” probability distribution function.

5. Comparing Probability Distribution Function (for discrete r
Comparing Probability Distribution Function (for discrete r.v) and Probability density function (for continuous r.v)

6. Comparing Probability Distribution Function (Discrete r
Comparing Probability Distribution Function (Discrete r.v) and Probability density function
Probability distribution function for a discrete random variable shows the probability for each outcome.

7. Comparing Probability Distribution Function (Discrete r
Comparing Probability Distribution Function (Discrete r.v) and Probability density function (Continuous r.v)
f(x) a b x
Probability Density Function shows the probability that the random variable falls in a particular range.

8. Probability density function
Let X be a continuous random variable. Let x denotes the value that the random variable X takes. We use f(x) to denote the probability density function.

9. Example
You client told you that he will visit you between noon and 2pm. Between noon and 2pm, the time he will arrive at your company is totally random. Let X be the random variable for the time he arrives (X=1.5 means he visit your office at 1:30pm)
Let x be the possible value for the random variable X. Then, the probability density function f(x) has the following shape.

10. Probability Density Function Example
f(x)
0.5 x 2

11. Probability Density Function Example
The probability that your client visits your office between 12:30 and 1:00 is given by the shaded area.
f(x)
0.5 1 x
0.5 2

12. Probability Density Function Example
Note that area between 0 and 2 should be equal to 1 since the probability that your clients arrives between noon and 2pm is 1 (assuming that he will keep his promise that he will visit between noon and 2pm)
f(x)
0.5 x 2

13. Some Properties of probability density function
f(x)≥0 for any x
Total area under f(x) is 1

14. Cumulative Distribution Function
The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x, as a function of x

15. In other words, the cumulative distribution function F(x) is given by the shaded area.
F(x)=P(X≤x) x x

16. Cumulative Distribution Function -Example-
f(x)
F(x)
Cumulative DistributionFunction
Density Function 1
0.5 x 2 2

17. A property of cumulative distribution functions

18. Probability Density Function and Cumulative Distribution Function -Exercise-
The daily revenue of a certain shop is a continuous random variable. Let X be the random variable for the daily revenue (in thousand dollars). Suppose that X has the following probability density function.
Ex.1 Graph f(x), Ex. 2 Find F(3),
Ex.3 Find P(2<X<3) Ex. 4 Graph F(x)

19. Relationship Between Probability Density Function and Cumulative Distribution Function
Let X be a continuous random variable. Then, there is a following relationship between probability density function and cumulative distribution function.

20. Expectation for continuous random variable
Expectation for a continuous random variable is defined in a similar way as the expectation for a discrete random variable.
See Next Page

21. Expectation for the discrete random variable is defined as
For continuous variable, f(x) corresponds to P(x). However, we cannot sum xf(x) since the value of x is continuous. Therefore, we define the expectation using integral in the following way.
where is the lowest possible value of X and is the highest possible value for X. We use μX to denote E(X) .

22. Expectation of a function of a continuous random variable
Let g(X) be a function of a continuous random variable X. Then the expectation of g(X) is given by the following.
A special case is the variance given in the next slide

23. Variance and standard deviation for the continuous random variable
Variance and standard deviation for a continuous random variable are defined as

24. Expected value for Continuous Random variable -Exercise-
Continue using the daily revenue example. The probability density function for X is given by
Compute E(X) and SD(X)

25. Linear Functions of Variables
Let X be a continuous random variable with mean X and variance X2, and let a and b any constant fixed numbers. Define the random variable W as
Then the mean and variance of W are
and
and the standard deviation of W is

26. Linear Functions of Variables -Exercise-
Continue using the daily sales example.
Suppose that the manager’s daily salary in thousand dollars is determined in the following way.
W=4+0.5X
Ex: Compute E(W) and SD(W)

27. Linear Functions of Variable
An important special case of the previous results is the standardized random variable
which has a mean 0 and variance 1.

28. Normal Distribution
A random variable X is said to be a normal random variable with mean μ and variance σ2 if X has the following probability density function.
where e and  are physical constants, e = and  =

29. Probability Density Function for a Normal Distribution
0.4
0.3
0.2
0.1
0.0  x

30. A note about the normal distribution
Normal distributions approximate many types of distributions.
Normal distributions have been applied to many areas of studies, such as economics, finance, operations management etc.

31. Normal Distribution approximates many types of distributions.

32. The shape of the normal distribution
The shape of the normal density function changes with the mean and the standard deviation

33. The shape of normal distribution. Exercise
Open “Normal Density Function Exercise”.
Excel function NORMDIST(x, μ,σ,FALSE) will compute the normal density function with mean μ and standard deviation σ evaluated at x.
EX1: Plot the normal density functions for μ=0 and σ=1, and μ=1 and σ=1. You will see how the change in the mean affect the shape of the density (without changing standard deviation)
EX2: Plot the normal density function for μ=0 and σ=1, and μ=0 and σ=2. You will see how the change in the standard deviation affects the shape (without changing the mean)

34. Answer
Changing the mean without changing the standard deviation causes a shift.
Increasing standard deviation makes the shape flatter (Fat Tail).

35. Properties of the Normal Distribution
Suppose that the random variable X follows a normal distribution given by the previous slides. Then
The mean of the random variable is ,
The variance of the random variable is 2,
The Following notation means that a random variable has the normal distribution with mean  and variance 2.

36. Cumulative Distribution Function of the Normal Distribution
Suppose that X is a normal random variable with mean  and variance 2 ; that is X~N(, 2). Then the cumulative distribution function is
This is the area under the normal probability density function to the left of x0, as illustrated in the figure in the next slide.

37. Shaded Area is the Probability that X does not Exceed x0 for a Normal Random Variable
f(x) x0 x

38. Range Probabilities for Normal Random Variables
Let X be a normal random variable with cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. Then
The probability is the area under the corresponding probability density function between a and b.

39. Range Probabilities for Normal Random Variables (Figure 6.12)
f(x) a b x 

40. The Standard Normal Distribution
The Standard Normal Distribution is the normal distribution with mean 0 and variance 1. We often use Z to denote the Standard Normal Variable.
If Z has standard normal distribution, we say that Z follows the standard normal distribution.

41. Computing the cumulative distribution and a range probability for the standard normal distribution
We often need to compute the cumulative distribution and a range probability for a standard normal distribution.
Since the standard normal distribution is used so frequently, most of the textbooks provide the cumulative distribution table. See P837 of our textbook.

42. Compute the following probability P(Z≤1.72) P(1.25<Z<1.72)
Computing the cumulative distribution and a range probability for the standard normal distribution -Exercise-
Compute the following probability
P(Z≤1.72)
P(1.25<Z<1.72)
P(Z≤-1.25)
P(-1.25<Z<1.72)

43. Finding Range Probabilities for Normally Distributed Random Variables
Let X be a normally distributed random variable with mean  and variance 2. Then the random variable Z = (X - )/ has a standard normal distribution: Z ~ N(0, 1)
It follows that if a and b are any numbers with a < b, then
where Z is the standard normal random variable and F(z) denotes its cumulative distribution function.

44. Computing Normal Probabilities (Example 6.6)
A very large group of students obtains test scores that are normally distributed with mean 60 and standard deviation 15. If you randomly choose a student, what is the probability that the test score of the student is between 85 and 95?

45. Joint Cumulative Distribution Functions
Let X1, X2, . . .Xk be continuous random variables
Their joint cumulative distribution function, F(x1, x2, . . .xk) defines the probability that simultaneously X1 is less than x1, X2 is less than x2, and so on; that is
The cumulative distribution functions F1(x1), F2(x2), . . .,Fk(xk) of the individual random variables are called their marginal distribution functions. For any i, Fi(xi) is the probability that the random variable Xi does not exceed the specific value xi.
The random variables are independent if and only if

46. Joint Cumulative Distribution Functions
Joint cumulative distribution functions often have complex forms, especially when they are not independent.
Some simple joint distribution functions are presented in the next page.

47. Joint distribution functions -Example, optional-
X and Y are said to have independent joint uniform distribution if X and Y has the following distribution function.
F(x,y)=xy 0≤x≤1, 0≤y≤1
X and Y are said to have independent joint exponential distribution if the joint distribution function is given by
F(x,y)=(1-e-x)(1-e-y), <x<∞、 0<y<∞

48. Expectation of a function of two random variables.
Let g(X,Y) be a function of two continuous random variables X and Y. Let f(x,y) be the joint distribution function of X and Y. Then the expectation of g(X,Y) is defined as
Special case is the covariance in the next slide

49. Covariance
Let X and Y be a pair of continuous random variables, with respective means x and y. The expected value of (x - x)(Y - y) is called the covariance between X and Y. That is
An alternative but equivalent expression can be derived as
If the random variables X and Y are independent, then the covariance between them is 0. However, the converse is not true.

50. Correlation
Let X and Y be jointly distributed random variables. The correlation between X and Y is

51. Sum of two random variables
Let X and Y be jointly distributed random variables. Let μX be the mean of X, μY be the mean of Y, σ2X be the variance of X ,σ2Y be the variance of Y. Then, we have
E(aX+bY) = aμX + bμY
Var(aX+bY) = a2 σX2 +b2 σY2 +2abCov(X,Y)
= a2 σX2 +b2 σY2 +2abCorr(X,Y)σXσY

52. Sums of Random Variables -Exercise-
Let X be the return for the stock A, and Y be the return for stock B. You have some fixed amount of money to invest in Stock A and Stock B. Let a be the share of money invested in stock A. (if you invest half of the money in stock A, a=0.5). Let b be the share of money invested in stock B (that is b=1-a). Then the return from the portfolio W is given by
Return of the portfolio: W=aX+bY

53. Suppose that μX=0.4, μY=0.1, σ2X=0.5, σ2Y=0.2, Corr(X,Y)=-0.2
Ex 1.
Suppose you invest 40% of your money in stock A, (i.e., a=0.4). Let W be the return of this portfolio. Compute E(W) and Var(W), and SD(W).
Ex 2.
Compute E(W) and SD(W) for different values of a, using “sum of random variables exercise”. Plot the relationship between SD(W) and E(W).

54. Answer for Ex2