Random Variables Probability Distributions Random Variables Probability Distributions
Random Variables. Probability Distributions
Random Variables. Probability Distributions • In slide 1 we considered frequency distribution of data. • These distributions show the absolute or relative frequency of the data values. • Similarly, a probability distribution or, briefly, a distribution, shows the probabilities of events in an experiment. • The quantity that we observe in an experiment will be denote by X and called a random variable (or stochastic variable) because the value it will assume in the next trial depends, on chance, or randomness.
Random Variables. Probability Distributions • If you roll a die, you get one of the numbers from 1 to 6, but you don’t know which one will show up next. • Thus X = Number o die turns up is a random variable. • So is X = Elasticity of rubber (elogation at break). • ‘Stochastic’ means related to chance.
Random Variables. Probability Distributions • If we count (cars on road, defecetive screwa in a production, tosses until a die shows the first Six), we have a discrete random variable and distibution. • If we measure (electirc voltage, rainfall, hardness of steel), we have a continuous random variable and distibution. • Precise definitions follow. In both cases distributionof X is determined by the distibution function • This is the probability that in trial, X will assume any value not exceeding x.
Caution ! • The terminology is not uniform. • F(x) is sometimes also called the cumulative distribution function.
Random Variable •
Random Variable •
Random Variable •
Random Variable •
Discrete Random Variables and Distributions •
Discrete Random Variables and Distributions • Clearly, the distribution of X also determined by the probability function f(x) or X, defined by • From this we get the values of the distibution function F(x) by taking sums,
Discrete Random Variables and Distributions •
Example • X = Number a fair die turns up. • Show the probability function f(x) and the distibution function F(x) of the discrete random variable • X has the possible values x = 1, 2, 3, 4, 5, 6 with the probability 1/6 each. • At these x distribution function has upward jumps of magnitude 1/6. • Hence from the graph of f(x) we can constract the graph of F(x) and conversly.
Example Probability Function Distrubution Function
Example •
Example
Example • X= sum of the two numbers obtained in tossing two fair dice once Probability Function Distrubution Function
Discrete Random Variables and Distributions • Two useful formulas for discrete distributions are readily obtained as follows. • For the probability corresponding to intervals we have (2) and (4)
Discrete Random Variables and Distributions •
Example •
Waiting Time Problem (Countably Infinite Sample Space) • In tossing a fair coin, let X = Number of trials until the first head appears. • Then, by independence of events
Waiting Time Problem (Countably Infinite Sample Space) •
Continuos Random Variables and Distributions • Discrete random variables appear in experiments in which we count (defectives in a production, days of sunshine in Eskişehir, customer standing in a line, etc. ). • Continuous random variables appear in experiments in which we measure (length of screws, voltage in a power line, etc. ).
Continuos Random Variables and Distributions • By definition, a random variable X and its distibution are of continuous type or, briefly, comtinuous, i its distribution function F(x) [defined in (1)] can be given by an ntegral.
Continuos Random Variables and Distributions •
Continuos Random Variables and Distributions • From (2) and (7) we obtain the very important formula for the probability corresponding to an interval.
Continuos Random Variables and Distributions • This is the analog of (5). • From (7) and P(S)=1 we also have the analog of (6):
Continuos Random Variables and Distributions •
Continuos Random Variables and Distributions • Answer is : This probability is the area under the density curve, as in the figüre and does not change by adding or subtracting a single point in the interval of integration. This different from the discrete case. • The next example illustrates notations and typical applications of our present formula (9).
Example •
Example •
Example • and • Note that the upper limit of integration is 1, not 2. Finally: • Algebraic simplification gives 3 x – x 3 = 1. 8. A solution is x=0. 73, approximetely.
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