PROBABILITY AND STATISTICS WEEK 5 Onur Doan 2016

PROBABILITY AND STATISTICS WEEK 5 Onur Doğan 2016 -2017

Random Variable. Let S be the sample space for an experiment. A real-valued function that is defined on S is called a random variable. Onur Doğan 2016 -2017

Random Variable 1. Discrete Random Variable: Has a finite (or countably infinite) range. • Tossing a coin: X= 0 for head and X= 1 for tail 2. Continuous Random Variable: Has an interval of real numbers for its infinite range. • The life length of a light bulb: X ≥ 0 Onur Doğan 2016 -2017

Reminder ! 1. 2. 3. is the mean of the sample s 2 and s are the variance and standard deviation of the sample , s 2, and s are called sample statistics 4. m (lowercase Greek letter “mu”) is the mean of the population 5. s 2 (“sigma squared”) is the variance of the population 6. s (lowercase Greek letter “sigma”) is the standard deviation of the population 7. m, s 2, and s are called population parameters. (A parameter is a constant. m, s 2, and s are typically unknown values. ) Onur Doğan 2016 -2017

Discrete Random Variables Let 4 coins tossed, and let X be the number of heads that are obtained. Let us find the distributions of that experiment. Onur Doğan 2016 -2017

Probability Distribution Onur Doğan 2016 -2017

Discrete Random Variables Example Three balls, a, b, c, are randomly distributed in three boxes. Determine the distribution of the random variable X ="the number of non-empty boxes". Onur Doğan 2016 -2017

Discrete Random Variables Example Consider a group of five potential blood donors; “a, b, c, d, and e” of whom only a and b have type 0+ blood. Five blood samples, one from each individual, will be typed in random order until an 0+ individual is identified. Let the rv Y=“the number of typings necessary to identify an 0+ individual. ” Find the pmf. Onur Doğan 2016 -2017

The Cumulative Distribution Function Onur Doğan 2016 -2017

Example Onur Doğan 2016 -2017

The Expected Value of X (Mean of a Discrete Random Variable) Onur Doğan 2016 -2017

The Expected Value of X (Mean of a Discrete Random Variable) • The mean, m, of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together: m = å [ x. p ( x )] Notes: n The mean is the average value of the random variable, what happens on average n The mean is not necessarily a value of the random variable Onur Doğan 2016 -2017

The Variance of X Onur Doğan 2016 -2017

Example The grades of n = 50 students in a statistics class are summarized as follows: Grade (X) Number of Students 1 2 3 4 10 20 15 5 Find the pmf, mean, variance and sd. Onur Doğan 2016 -2017

A Shortcut Formula for V(X) Onur Doğan 2016 -2017

Example Suppose that a random variable X can take each of the five values − 2, 0, 1, 3, and 4 with equal probability. Determine the variance and standard deviation of X. Onur Doğan 2016 -2017

Example Determine the mean, variance, and standard deviation of casting a single die (X). Onur Doğan 2016 -2017

Example Onur Doğan 2016 -2017

Example A shipment of 8 similar microcomputers to contains 3 defective one. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives. Onur Doğan 2016 -2017

Example Onur Doğan 2016 -2017

Example: The probability distribution for a random variable x is given by the probability function: Find the mean, variance, and standard deviation Onur Doğan 2016 -2017