Lecture 10 Mathematical Expectation Mathematical Expectation The expected

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Lecture (10) Mathematical Expectation

Lecture (10) Mathematical Expectation

Mathematical Expectation The expected value of a variable is the value of a descriptor

Mathematical Expectation The expected value of a variable is the value of a descriptor when averaged over a large number theoretically infinite.

Mathematical Expectation (cont. ) Another way to compute the variance

Mathematical Expectation (cont. ) Another way to compute the variance

Example 1 Sample Space TT TH Number of Heads 0 1 HT HH 2

Example 1 Sample Space TT TH Number of Heads 0 1 HT HH 2

Example 1 (cont. )

Example 1 (cont. )

Example 1 (cont. ) Experiment: Toss Two Coins PROBABILITY 1 0. 5 . 25

Example 1 (cont. ) Experiment: Toss Two Coins PROBABILITY 1 0. 5 . 25 0 1 2 NUMBER OF HEADS 3

Example 1 (cont) • E. G. Toss 2 coins, count heads, compute expected value:

Example 1 (cont) • E. G. Toss 2 coins, count heads, compute expected value: • = 0 . 25 + 1 . 50 + 2 . 25 = 1 E. G. Toss 2 coins, count heads, compute variance: variance = (0 - 1)2 (. 25) + (1 - 1)2 (. 50) + (2 - 1) 2(. 25) =. 50

Example 2

Example 2

Discrete Uniform Distribution Example • Find the mean of the number of spots that

Discrete Uniform Distribution Example • Find the mean of the number of spots that appear when a die is tossed. The probability distribution is given below.

Discrete Uniform Distribution Example (cont. ) That is, when a die is tossed many

Discrete Uniform Distribution Example (cont. ) That is, when a die is tossed many times, theoretical mean will be 3. 5.

Binomial Distribution - Example • A coin is tossed four times. Find the mean,

Binomial Distribution - Example • A coin is tossed four times. Find the mean, variance and standard deviation of the number of heads that will be obtained. • Solution: n = 4, p = 1/2 and q = 1/2. • = n p = (4)(1/2) = 2. • 2 = n p q = (4)(1/2) = 1. • = = 1.

Poisson Distribution

Poisson Distribution

Uniform Distribution Example

Uniform Distribution Example

Example If the probability density function has the form f(x) = ax for a

Example If the probability density function has the form f(x) = ax for a random variable X between 0 and 2. (a) Find the value of a. (b) Find the median of X (c) Find P(1. 0 < X < 2. 0) Solution: (a) From the area under the whole density curve is 1, then we have

Quiz

Quiz

Exponential Distribution

Exponential Distribution

Comparison of Parameters of Dist’n Distribution Normal Log. N x Y =logx Mean x

Comparison of Parameters of Dist’n Distribution Normal Log. N x Y =logx Mean x y Gamma x nk Exp t 1/k Variance s x 2 s y 2 nk 2 1/k 2 Skewness zero 2/n 0. 5 2