1. 2.1 Introduction In an experiment of chance, outcomes occur randomly. We often summarize the outcome from a random experiment by a simple number. Definition 2.1 A variable is a symbol such as X, Y, Z, x or H, that assumes values for different elements. If the variable can assume only one value, it is called a constant. A random variable is a variable whose value is determined by the outcome of a random experiment.

2. Example 2.1 A balanced coin is tossed two times. List the elements of the sample space, the corresponding probabilities and the corresponding values x, where X is the number of getting head. Solution Also we can write P(X=2)=1/4, for example, for the probability of the event that the random variable X will take on the value 2. Elements of sample space Probabilityx HH¼2 HT¼1 TH¼1 TT¼0

3. Definition 2.2: Discrete Random Variables A random variable is discrete if its set of possible values consist of discrete points on the number line. Continuous Random Variables A random variable is continuous if its set of possible values consist of an entire interval on the number line.

4. Examples of discrete random variables: number of scratches on a surface proportion of defective parts among 1000 tested number of transmitted bits received error Examples of continuous random variables: electrical current length pressure time voltage

5. Definition 2.3: If X is a discrete random variable, the function given by f(x)=P(X=x) for each x within the range of X is called the probability distribution of X. Requirements for a discrete probability distribution:

6. Example 2.2 Check whether the distribution is a probability distribution. Solution # so the distribution is not a probability distribution. X01234 P(X=x)0.1250.3750.0250.3750.125

7. Example 2.3 Check whether the function given by can serve as the probability distribution of a discrete random variable.

8. Solution # so the given function is a probability distribution of a discrete random variable.

9. Definition 2.4: A function with values f(x), defined over the set of all numbers, is called a probability density function of the continuous random variable X if and only if

10. Requirements for a probability density function of a continuous random variable X:

11. Example 2.4 Let X be a continuous random variable with the following probability density function

12. Solution a)

13. b)

14.  The cumulative distribution function of a discrete random variable X, denoted as F(x), is  For a discrete random variable X, F(x) satisfies the following properties:  If the range of a random variable X consists of the values

15.  The cumulative distribution function of a continuous random variable X is

16. Example 2.5 solution x1234 f(x) 4/10 3/102/101/10 F(x)4/107/109/101

17. Example 2.6 If X has the probability density

18. Solution

21. 2.5.1 Expected Value  The mean of a random variable X is also known as the expected value of X as  If X is a discrete random variable,  If X is a continuous random variable,

23. 2.5.4 Properties of Expected Values For any constant a and b,

24. 2.5.5 Properties of Variances For any constant a and b,

25. Example 2.7 Find the mean, variance and standard deviation of the probability function

26. Solution Mean:

27. Varians:

28. Example 2.8 Let X be a continuous random variable with the Following probability density function

29. Solution