Understanding Random Variables and Distribution Functions in Probability Theory
Learn about random variables, sample spaces, probability mass functions, and more in the context of probability theory through examples and definitions. Explore how random variables are classified and how distribution functions are utilized in both discrete and continuous scenarios.
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Random Random Variables AND Variables AND DISTRIBUTION DISTRIBUTION FUNCTION FUNCTION
Considerthe 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. Solution: S={ HH , HT , TH , TT } 2 1 1 0
Definition (1): A random variable is a function that associates a real number with each element in the sample space. Remark: 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 values
Definition 3.3 If the space of random variable X is countable, then X is called a discrete random variable. Definition 3.4 If the space of random variable X is uncountable, then X is called a continuous random variable.
3.2. Distribution Functions of Discrete Random Variables Definition 3.5. Let ?? be the space of the random variable X. The function f : ?? f(x) = P(X = x) is called the probability mass function (p m f) of X. IR defined by
Example 3.5 A pair of dice consisting of a six-sided die and a four-sided die 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.
Answer: The sample space of this random experiment is given by {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)} S =
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 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) = ? ??
Example 3.6. A fair coin is tossed 3 times. Let the random variable X denote 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. Answer: The sample space S of this experiment consists of all binary sequences of length 3, that is S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}.
The space of this random variable is given by ??= {0, 1, 2, 3}. Therefore, the probability density function of X is given by f(0) = P(X = 0) =? f(1) = P(X = 1) =3 8 8 f(2) = P(X = 2) =3 8 f(3) = P(X = 3) =? 8
Theorem 3.1 If X is a discrete random variable with space ?? and probability density function f(x), then (a). f(x) 0 for all x in, and (b). f(x) = 1. Example 3.7 If the probability of a random variable X with space ??= {1, 2, 3, ..., 12} is given by f(x) = k (2x 1), then, what is the value of the constant k?
Answer: 1 = f(x) 12k (2x 1) 1 = 1 12(2x 1) 1 = k 1 1= k 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 1 = k 144. K= 144 1
Definition 3.6. The cumulative distribution function F(x) of a random variable X is defined as F(x) = P(X=x) for all real numbers x. Theorem 3.2. If X is a random variable with the space??, then F(X) = ? ??(?)
Example 3.8 If the probability density function of the random variable X is given by 1 144(2x 1) for x = 1, 2, 3, ..., 12 then find the cumulative distribution function of X. Answer: The space of the random variable X is given by ??= {1, 2, 3, ..., 12}.
Then ? F(1) = ? 1?(?)= f(1) = 144 4 ? 3 F(2) = ? 2?(?)= f(1) + f(2) = 14?+ 144= 144 ? 3 5 9 F(3) = ? 3?(?) = f(1) + f(2) + f(3) = .. ........ .. ........ 144+ 144+ 144= 144 F(12) = ? 12?(?) = f(1) + f(2) + + f(12) = 1.
This part is in the book from pg 45 best of luck
