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Lecture 3 B Maysaa ELmahi 3.3. Distribution Functions of Continuous Random Variables Recall that a random variable X is said to be continuous if its space is either an interval or a union of intervals. Definition 3.7. Let X be a continuous random variable whose space is the set of real numbers I R. A nonnegative real valued function f : IR IR is said to be the probability density function for the continuous random variable X if it satisfies: (a) โ ๐ โโ ๐ฑ ๐๐ฑ = ๐ , and (b) if A is an event, then ๐ฉ ๐ = ๐ ๐ ๐ฑ ๐๐ฑ Example 3.10. Is the real valued function f : IR ๐๐ฑ โ๐ IR defined by ๐ข๐ ๐ < ๐ฑ < ๐ ๐ ๐ฑ = ๐ ๐จ๐ญ๐ก๐๐ซ๐ฐ๐ข๐ฌ๐ (a) probability density function for some random variable X? Answer: โ ๐ ๐๐ฑ โ๐ ๐๐ฑ ๐ ๐ฑ ๐๐ฑ = โโ ๐ = โ๐ ๐ ๐ ๐ฑ ๐ โ๐ ๐ ๐ โ ๐ =1 Thus f is a probability density function. Example 3.11. Is the real valued function f : IR ๐+ ๐ฑ IR defined by ๐ข๐ โ ๐ < ๐ฑ < ๐ ๐ ๐ฑ = ๐ ๐จ๐ญ๐ก๐๐ซ๐ฐ๐ข๐ฌ๐ (a) probability density function for some random variable X? Answer: โ 1 ๐ ๐ฅ ๐๐ฅ = (1 + ๐ฅ ) ๐๐ฅ โโ โ1 0 = 1 1 โ ๐ฅ ๐๐ฅ + โ1 0 1 = ๐ฅ โ 2 ๐ฅ2 =1 + (1 + ๐ฅ) ๐๐ฅ 1 2 0 + โ1 1 1 ๐ฅ + 2 ๐ฅ2 +1+2 = 3 1 0 Definition 3.8. Let f(x) be the probability density function of a continuous random variable X. The cumulative distribution function F(x) of X is defined as ๐ ๐ฑ =๐ ๐โค๐ฑ = ๐ฑ ๐ โโ ๐ญ ๐๐ญ Theorem 3.5. If F(x) is the cumulative distribution function of a continuous random variable X, the probability density function f(x) of X is the derivative of F(x), that is ๐ ๐ ๐ฑ = ๐(๐ฑ) ๐๐ฑ Theorem 3.6. Let X be a continuous random variable whose c d f is F(x). Then followings are true: ๐ . ๐ ๐ < ๐ฅ = ๐น(๐ฅ) ๐ . ๐ ๐ > ๐ฅ = 1 โ ๐น(๐ฅ) ๐ .๐ ๐ = ๐ฅ = 0 ๐ .๐ ๐ < ๐ < ๐ = ๐น(๐) โ ๐น(๐) Example : (H.W) ๐ + 1 ๐ฅ2 ๐๐ 0<๐ฅ<1 ๐ ๐ฅ = 0 ๐๐กโ๐๐๐ค๐๐ ๐ a. what is the value of the constant k? b. What is the probability of X between the first and third? d. What is the cumulative distribution function? 4.2. Expected Value of Random Variables Definition 4.2. Let X be a random variable with space ๐ ๐ฅ and probability density function f(x). The mean ๐๐ฅ of the random variable X is defined as ๐๐(๐) ๐๐ X is discrete ๐โ๐น๐ฟ ๐๐ฑ = ๐(๐ฑ) = โ ๐ ๐ ๐ ๐ ๐ โโ if X is continuous Example : x 0 1 2 3 P(x) 1/8 3/8 3/8 1/8 what is the mean of X? Answer: ๏ญ ๏ฝ E (x ) ๏ฝ ๏ฅ x f (x ) x = 0* 1/8+ 1* 3/8 +2* 3/8 +3 * 1/8 = 0 + 3/8 + 6/8 + 3/8 = 12/8 Example : ๐ ๐ ๐ ๐ = ๐ ๐๐ ๐<๐<๐ ๐๐๐๐๐๐๐๐๐ Answer: ๏ฅ ๏ญ ๏ฝ E (x ) ๏ฝ ๏ฒ x f (x )dx ๏ญ๏ฅ 7 = 2 1 ๐ฅ ๐๐ฅ 5 = 1 2 7 ๐ฅ 2 10 = 1 10 45 9 49 โ 4 = 10 =2 Theorem 4.1 Let X be a random variable with p d f f(x). If a and b are any two real numbers, then ๐. ๐(๐๐ฑ + ๐ ) = a ๐ ๐ฑ + ๐ b. ๐(๐๐ฑ) = ๐ ๐(๐ฑ) c. ๐(๐) = ๐ 4.3. Variance of Random Variables Definition 4.4. Let X be a random variable with mean ๐๐ฅ . The variance of X, denoted by Var(X), is defined as ๐๐๐ซ ๐ฑ = ( ๐ ๐ฑ โ ๐๐ฑ ) ๐ ๐๐ ๐ฑ = ๐ ๐ฑ ๐ โ (๐๐ ๐ฑ ) Example : 2(๐ฅ โ 1) ๐๐ 1<๐ฅ<2 ๐ ๐ฅ = 0 ๐๐กโ๐๐๐ค๐๐ ๐ a. what is the variance of X? Answer: โ ๐ฅ โโ ๐๐ฅ = ๐ธ ๐ฅ = ๐ ๐ฅ ๐๐ฅ = =2 2 2 (๐ฅ โ๐ฅ)๐๐ฅ 1 =2 ๐ฅ3 3 โ =2 8 3 โ2 =2โ 2(๐ฅ โ 1)๐๐ฅ ๐ฅ2 2 2 1 4 5 6 2 ๐ฅ 1 = โ 10 6 1 3 1 4 1 โ 2 = 2((6 - ( - 6 ) ) โ ๐ธ ๐ฅ 2 2 2 = ๐ฅ 2 2(๐ฅ โ 1)๐๐ฅ ๐ฅ ๐ ๐ฅ ๐๐ฅ = โโ 1 = 2 2 3 2 (๐ฅ โ(๐ฅ )๐๐ฅ 1 =2 ๐ฅ4 4 โ =2 16 4 โ3 ๐ฅ3 2 3 1 8 17 โ 1 4 1 16 1 โ 3 = 2((12 - ( - 12 ) ) = 2((12) ) = 17/6 Thus, the variance of X is given by ๐๐ ๐ฑ = ๐ ๐ฑ ๐ โ (๐๐ ๐ฑ ) = 17 6 โ 10 6 = Remark: Var (๐๐ฅ + ๐ ) = ๐2 ๐๐๐ ๐ฅ ๐๐๐ (๐ ๐ฅ ) =๐2 ๐๐๐ ๐ฅ ๐๐๐ (๐) =0 7 6 4.1. Moments of Random Variables Definition 4.1. The nth moment about the origin of a random variable X, as denoted by E(๐ฅ ๐ ), is defined to be ๐ฑ ๐ง ๐(๐ฑ) ๐ข๐ X is discrete ๐ฑโ๐ ๐ ๐ ๐ฑ๐ง = โ ๐ฑ ๐ง ๐ ๐ฑ ๐๐ฑ โโ if X is continuous for n = 0, 1, 2, 3, ...., provided the right side converges absolutely. If n = 1, then E(X) is called the first moment about the origin. If n = 2, then E(๐ฅ 2 ) is called the second moment of X about the origin. 4.5. Moment Generating Functions Definition 4.5. Let X be a random variable whose probability density function is f(x). A real valued function M : IR IR defined by ๐ด ๐ = ๐ฌ(๐๐๐ ) is called the moment generating function of X if this expected value exists for all t in the interval โh < t < h for some h > 0. Using the definition of expected value of a random variable, we obtain the explicit representation for M(t) as ๐๐ญ๐ฑ ๐(๐ฑ) ๐ข๐ X is discrete ๐ฑโ๐ ๐ ๐(๐ญ) = โ ๐๐ญ๐ฑ ๐ ๐ฑ ๐๐ฑ โโ if X is continuous