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Random Variables and
Probability Distributions
• capital letter X, to denote a random variable
• corresponding small letter, x for one of its values.
• Statistics is concerned with making inferences about
populations and population characteristics.
• the assignment of 1 or 0 is arbitrary though quite convenient. • The random variable for which 0 and 1 are chosen• to describe the two possible values is called a Bernoulli random variable.
• a statistical experiment, any process by which several chance observations are generated
• S = {NNN, NND, NDN, DNN, NDD, DND, DDN, DDD},where N nondefective and D defective.
• concerned with the number of defectives• A random variable is a function that associates a real
number with each element in the sample space. • the random variable X assumes the value 2 • for all elements in the subset E = {DDN, DND, NDD} of
the sample space S.
• If a sample space contains a finite number of possibilities
• Or an unending sequence with as many elements as there are
whole numbers,
• it is called a discrete sample space.
• If a sample space contains an infinite number of possibilities
• equal to the number of points on a line segment,
• it is called a continuous sample space.
• A random variable is called a discrete random variable
• if its set of possible outcomes is countable.
• But a random variable whose set of possible values is an
entire interval of numbers is not discrete.
• When a random variable can take on values on a continuous
scale,
• it is called a continuous random variable.
Let W be a random variable
in three tosses of a coin.
number of heads minus the number of tails
List the elements of the sample space S
to each sample point assign a value w of W.
What is a cumulative probability distribution (CD)?
• A table of the probabilities
cumulated over the events.
• The CD is a monotonically
increasing set of numbers
• The CD always ends with at
the highest value of 1.
Examples of e probability distribution (PD): Bernoulli Random Variables (RV)
•Bernoulli distribution: The outcome is either a
• “failure” (0) or a (1).
•X is a bernoulli RV when
•Pr(X = 0) = p “failure”
•Pr(X = 1) = (1 − p) “success”
• The bernoulli distribution has one parameter, p
The cumulative probability distribution (CD)of a Bernoulli Random Variables (RV)
• We need to know what p of
the RV is. Say p = 0.35.
• The CD of this Bernoulli RV
is:
• X is a random variable whose values x are the possible numbers of defective computers purchased by the school.
• Then x can only take the numbers 0, 1, 2
It is often helpful to look at a probability distribution in graphic form.
Continuous Probability Distributions
• A probability density function is constructed
• the area under its curveis 1
The CDF, F(x), is area function of the PDF, obtained by integrating the PDFfrom negative infinity to an arbitrary value x.
Joint Probability Distributions
• X and Y are two discrete random variables, • the probability distribution for their simultaneous
occurrence by a function f(x, y)• pair of values (x, y) within the range of the random
variables X and Y .• refer to this function as the joint probability distribution of
X and Y .• in the discrete case, f(x, y) = P(X = x, Y = y);• f(x, y) give the probability that outcomes x and y occur at
the same time.
Joint Probability Distributions
• Sometimes we’re simultaneously interested in two or more variables in a random experiment.
• We’re looking for a relationship between the two variables.
• Examples for discrete r.v.’s • • Year in college vs. Number of credits taken
• Number of cigarettes smoked per day vs. Day of the week
• Examples for continuous r.v.’s• Time when bus driver picks you up vs. Quantity of caffeine in bus driver’s
system
• Dosage of a drug (ml) vs. Blood compound measure (percentage)
• probability distribution g(x) of X alone is obtained by summing f(x, y) over the values of Y .
• probability distribution h(y) of Y alone is obtained by summing f(x, y) over the values of X.
• g(x) and h(y) to be the marginal distributions of X and Y
we rolled two dice and let X be the value on the first die and T be the total on both dice. Compute the marginal pmf of X and of T.
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