Consider the experiment of tossing a coin twice If we are interested in the number of heads that show on the top face describe the sample space S HH HT TH TT 2 1 1 0 ID: 1028278
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1. Random Variables AND DISTRIBUTION FUNCTION
2. Consider the experiment of tossing a coin twice. If we are interested in the number of heads that show on the top face, describe the sample space.S={ HH , HT , TH , TT } 2 1 1 0Solution:
3. A random variable is a function that associates a real number with each element in the sample space.Definition (1):We shall use a capital letter, say X, to denote a random variable and its corresponding small letter, x in this case, for one of its valuesRemark:
4. If the space of random variable X is uncountable, then Xis called a continuous random variable.If the space of random variable X is countable, then X iscalled a discrete random variable.Definition 3.4Definition 3.3
5. Let be the space of the random variable X. Thefunction f : IR defined byf(x) = P(X = x) is called the probability mass function (p m f) of X. 3.2. Distribution Functions of Discrete Random VariablesDefinition 3.5.
6. A pair of dice consisting of a six-sided die and a four-sideddie is rolled and the sum is determined. Let the random variable X denote this sum.Find the sample space, the space of the random variable, And probability mass function of X.Example 3.5
7. S = Answer:The sample space of this random experiment is given by
8. f(2) = P(X = 2) = , f(3) = P(X = 3) = , f(4) = P(X = 4) = f(5) = P(X = 5) = , f(6) = P(X = 6) = , f(7) = P(X = 7) = f(8) = P(X = 8) = , f(9) = P(X = 9) = , f(10) = P(X = 10) = The space of the random variable X is given by= {2, 3, 4, 5, 6, 7, 8, 9, 10}.Therefore, the probability mass function of X is given by
9. A fair coin is tossed 3 times. Let the random variable Xdenote the number of heads in 3 tosses of the coin. Find the sample space,the space of the random variable, and the probability density function of X.Example 3.6.The sample space S of this experiment consists of all binary sequencesof length 3, that isS = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}.Answer:
10. Therefore, the probability density function of X is given byf(0) = P(X = 0) = f(1) = P(X = 1) = f(2) = P(X = 2) = f(3) = P(X = 3) = The space of this random variable is given by= {0, 1, 2, 3}.
11. If X is a discrete random variable with space and probabilitydensity function f(x), then(a). f(x) 0 for all x in, and(b). = 1. Theorem 3.1If the probability of a random variable X with space = {1, 2, 3, ..., 12} is given byf(x) = k (2x − 1),then, what is the value of the constant k? Example 3.7
12. 1 = 1 = 1 = k1= k1 = k 144.K= Answer:
13. The cumulative distribution function F(x) of a randomvariable X is defined asF(x) = P(X=x)for all real numbers x.Definition 3.6.If X is a random variable with the space, thenF(X) = Theorem 3.2.
14. If the probability density function of the random variable Xis given by(2x − 1) for x = 1, 2, 3, ..., 12 then find the cumulative distribution function of X. Example 3.8The space of the random variable X is given by= {1, 2, 3, ..., 12}. Answer:
15. F(1) = = f(1) = F(2) = = f(1) + f(2) = + =F(3) = = f(1) + f(2) + f(3) = + + =.. .......... ........F(12) = f(1) + f(2) + · · · + f(12) = 1. Then
16. This part is in the book from pg 45 best of luck