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) −∞ ∞ 𝐟 𝐱 𝐝𝐱=𝟏 , and (b) if A is an event, then 𝐩 𝐀 = 𝐀 𝐟 𝐱 𝐝𝐱

Example 3.10. Is the real valued function f : IR IR defined by 𝐟 𝐱 = 𝟐𝐱 −𝟐 𝐢𝐟 𝟏<𝐱<𝟐 𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞 (a) probability density function for some random variable X? Answer: −∞ ∞ 𝐟 𝐱 𝐝𝐱= 𝟏 𝟐 𝟐𝐱 −𝟐 𝐝𝐱 =−𝟐 𝟏 𝐱 𝟐 𝟏

−𝟐 𝟏 𝟐 −𝟏 =1 Thus f is a probability density function. Example 3.11. Is the real valued function f : IR IR defined by 𝐟 𝐱 = 𝟏+ 𝐱 𝐢𝐟 −𝟏<𝐱<𝟏 𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞 (a) probability density function for some random variable X?

Answer: −∞ ∞ 𝑓 𝑥 𝑑𝑥= −1 1 (1+ 𝑥 ) 𝑑𝑥 = −1 0 1−𝑥 𝑑𝑥+ 0 1 (1+𝑥) 𝑑𝑥 = 𝑥− 1 2 𝑥 2 0 −1 + 𝑥+ 1 2 𝑥 2 1 0 =1 + 1 2 + 1 + 1 2 = 3

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 𝐅 𝐱 =𝐏 𝐗≤𝐱 = −∞ 𝐱 𝐟 𝐭 𝐝𝐭 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 𝐝 𝐝𝐱 𝐅 𝐱 =𝐟(𝐱)

Theorem 3.6. Let X be a continuous random variable whose c d f is F(x). Then followings are true: 𝑎 . 𝑃 𝑋<𝑥 =𝐹(𝑥) 𝑏 . 𝑃 𝑋>𝑥 =1− 𝐹(𝑥) 𝑐 . 𝑃 𝑋=𝑥 =0 𝑑 . 𝑃 𝑎<𝑋<𝑏 = 𝐹(𝑏) − 𝐹(𝑎)

Example : (H.W) 𝑓 𝑥 = 𝑘+1 𝑥 2 𝑖𝑓 0<𝑥<1 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 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 𝑅 𝑥 and probability density function f(x). The mean 𝜇 𝑥 of the random variable X is defined as 𝛍 𝐱 =𝐄(𝐱) = 𝒙∈ 𝑹 𝑿 𝒙𝒇(𝒙) 𝒊𝒇 X is discrete −∞ ∞ 𝒙 𝒇 𝒙 𝒅𝒙 if X is continuous

Example : x 1 2 3 P(x) 1/8 3/8 what is the mean of X? Answer: = 0* 1/8+ 1* 3/8 +2* 3/8 +3 * 1/8 = 0 + 3/8 + 6/8 + 3/8 = 12/8

Example : 𝒇 𝒙 = 𝟏 𝟓 𝒊𝒇 𝟐<𝒙<𝟕 𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆 Answer: = 2 7 𝑥 1 5 𝑑𝑥 = 1 10 𝑥 2 7 2 = 1 10 49−4 = 45 10 = 9 2

4.3. Variance of Random Variables 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 𝐚. 𝐄(𝐚𝐱+𝐛 ) = a 𝐄 𝐱 +𝐛 b. 𝐄(𝐚𝐱) = 𝐚 𝐄(𝐱) c. 𝐄(𝐚) = 𝐚 4.3. Variance of Random Variables Definition 4.4. Let X be a random variable with mean 𝜇 𝑥 . The variance of X, denoted by Var(X), is defined as

𝐕𝐚𝐫 𝐱 = ( 𝐄 𝐱 − 𝛍 𝐱 ) 𝟐 𝛔 𝟐 𝐱 =𝐄 𝐱 𝟐 − (𝛍 𝟐 𝐱 ) Example : 𝑓 𝑥 = 2(𝑥−1) 𝑖𝑓 1<𝑥<2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 a. what is the variance of X? Answer:

𝜇 𝑥 =𝐸 𝑥 = −∞ ∞ 𝑥 𝑓 𝑥 𝑑𝑥= 1 2 𝑥 2(𝑥−1)𝑑𝑥 = 2 1 2 (𝑥 2 −𝑥)𝑑𝑥 = 2 𝑥 3 3 − 𝑥 2 2 2 1 = 2 8 3 − 4 2 − 1 3 − 1 2 = 2(( 4 6 - ( - 1 6 ) ) =2∗ 5 6 = 10 6

𝐸 𝑥 2 = −∞ ∞ 𝑥 2 𝑓 𝑥 𝑑𝑥= 1 2 𝑥 2 2(𝑥−1)𝑑𝑥 = 2 1 2 (𝑥 3 − (𝑥 2 )𝑑𝑥 = 2 𝑥 4 4 − 𝑥 3 3 2 1 = 2 16 4 − 8 3 − 1 4 − 1 3 = 2(( 16 12 - ( - 1 12 ) ) = 2(( 17 12 ) ) = 17/6

Thus, the variance of X is given by 𝛔 𝟐 𝐱 =𝐄 𝐱 𝟐 − (𝛍 𝟐 𝐱 ) = 17 6 – 10 6 = 7 6 Remark: Var (𝑎𝑥+𝑏 ) = 𝑎 2 𝑉𝑎𝑟 𝑥 𝑉𝑎𝑟 (𝑎 𝑥 ) = 𝑎 2 𝑉𝑎𝑟 𝑥 𝑉𝑎𝑟 (𝑎) =0

4.1. Moments of Random Variables Definition 4.1. The nth moment about the origin of a random variable X, as denoted by E( 𝑥 𝑛 ), is defined to be 𝐄 𝐱 𝐧 = 𝐱∈ 𝐑 𝐗 𝐱 𝐧 𝐟(𝐱) 𝐢𝐟 X is discrete −∞ ∞ 𝐱 𝐧 𝐟 𝐱 𝐝𝐱 if X is continuous

for n = 0, 1, 2, 3,. , provided the right side converges absolutely 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( 𝑥 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 𝑴 𝒕 =𝑬( 𝒆 𝒕𝒙 ) is called the moment generating function of X if this expected value exists for all t in the interval −h < 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 𝐌(𝐭) = 𝐱∈ 𝐑 𝐗 𝐞 𝐭𝐱 𝐟(𝐱) 𝐢𝐟 X is discrete −∞ ∞ 𝐞 𝐭𝐱 𝐟 𝐱 𝐝𝐱 if X is continuous