A random variable is a variable whose values are numerical outcomes of a random experiment. That is, we consider all the outcomes in a sample space S and then associate a number with each outcome

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• A random variable is a variable whose values are
  numerical outcomes of a random experiment. That
  is, we consider all the outcomes in a sample
  space S and then associate a number with each
  outcome
• Example Toss a fair coin 4 times and let
• Xthe number of Heads in the 4 tosses
• We write the so-called probability distribution
  of X as a list of the values X takes on along
  with the corresponding probabilities that X takes
  on those values.

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• The figure below (Fig. 4.6) and Example 4.23 show
  how to get the probability distribution of X.
  Each outcome has prob1/16 (HINT use the and
  rule to show this), and then use the or rule to
  show that P(X1) P(TTTH or TTHT or THTT or
  HTTT) etc)

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• There are two types of r.v.s discrete and
  continuous. A r.v. X is discrete if the number
  of values X takes on is finite (or countably
  infinite). In the case of any discrete X, its
  probability distribution is simply a list of its
  values along with the corresponding probabilities
  X takes on those values.
• Values of X x1 x2 xk
• P(X) p1 p2 pk
• NOTE each value of p is between 0 and 1 and all
  the values of p sum to 1. We display probability
  distributions for discrete r.v.s with so-called
  probability histograms. The next slide shows the
  probability histogram for X of Hs in 4 tosses
  of a fair coin.

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The next slide gives a similar example...
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• The probability distribution of a random
  variable X lists the values and their
  probabilities
• The probabilities pi must add up to 1.
• A basketball player shoots three free throws. The
  random variable X is the number of baskets
  successfully made. Suppose he is a 50 free throw
  shooter...

Value of X 0 1 2 3
Probability 1/8 3/8 3/8 1/8
HMM HHM MHM HMH MMM MMH MHH HHH
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• The probability of any event is the sum of the
  probabilities pi of the values of X that make up
  the event.
• A basketball player shoots three free throws. The
  random variable X is the number of baskets
  successfully made. Suppose he is a 50 free throw
  shooter.

What is the probability that the player
successfully makes at least two baskets (at
least two means two or more)? USE THE OR
RULE!
Value of X 0 1 2 3
Probability 1/8 3/8 3/8 1/8
HMM HHM MHM HMH MMM MMH MHH HHH
P(X2) P(X2) P(X3) 3/8 1/8 1/2
What is the probability that the player
successfully makes fewer than three baskets? USE
THE OR RULE HERE TOO...!
P(Xlt3) P(X0) P(X1) P(X2) 1/8 3/8
3/8 7/8 or P(Xlt3) 1 P(X3) 1 1/8
7/8 (THIS IS THE NOT RULE)
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• A continuous r.v. X takes its values in an
  interval of real numbers. The probability
  distribution of a continuous X is described by a
  density curve, whose values lie wholly above the
  horizontal axis, whose total area under the curve
  is 1, and where probabilities about X correspond
  to areas under the curve.

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• The first example is the random variable which
  randomly chooses a number between 0 and 1
  (perhaps using the spinner on page 253 go over
  Example 4.25). This r.v. is called the uniform
  random variable and has a density curve that is
  completely flat! Probabilities correspond to
  areas under the curve... see next slide for the
  computations...

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• A continuous random variable X takes all values
  in an interval.
• Example There is an infinity of numbers between
  0 and 1 (e.g., 0.001, 0.4, 0.0063876).
• How do we assign probabilities to events in an
  infinite sample space?
• We use density curves and compute probabilities
  for intervals.
• The probability of any event is the area under
  the density curve for the values of X that make
  up the event.

This is a uniform density curve for the variable
X.
The probability that X falls between 0.3 and 0.7
is the area under the density curve for that
interval (base x height for this density) P(0.3
X 0.7) (0.7 0.3)1 0.4
X
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• The probability of a single point is meaningless
  for a continuous random variable. Only intervals
  can have a non-zero probability, represented by
  the area under the density curve for that
  interval.

The probability of a single point is zero since
there is no area above a point! This makes the
following statement true
The probability of an interval is the same
whether boundary values are included or
excluded P(0 X 0.5) (0.5 0)1
0.5 P(0 lt X lt 0.5) (0.5 0)1 0.5 P(0 X
lt 0.5) (0.5 0)1 0.5
P(X lt 0.5 or X gt 0.8) P(X lt 0.5) P(X gt 0.8)
1 P(0.5 lt X lt 0.8) 0.7 (You may use either
the OR Rule or the NOT Rule...)
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• The other example of a continuous r.v. that weve
  already seen is the normal random variable. See
  the next slide for a reminder of how weve used
  the normal and how it relates to probabilities
  under the normal curve...
• Go over Example 4.26 in detail! We saw earlier
  that p-hat had a sampling distribution which was
  normal. Thus p-hat can be treated as a normal
  random variable we have shown that the mean of
  p-hat is p and the standard deviation of p-hat is
  sqrt(p(1-p)/n). Now use this information to do
  Ex. 4.26

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Continuous random variable and population
distribution
individuals with X such that x1 lt X lt x2
The shaded area under a density curve shows the
proportion, or , of individuals in a population
with values of X between x1 and x2.
Because the probability of drawing one individual
at random depends on the frequency of this type
of individual in the population, the probability
is also the shaded area under the curve.
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Mean of a random variable

• The mean x bar of a set of observations is their
  arithmetic average.
• The mean µ of a random variable X is a weighted
  average of the possible values of X, reflecting
  the fact that all outcomes might not be equally
  likely.

A basketball player shoots three free throws. The
random variable X is the number of baskets
successfully made (H).
Value of X 0 1 2 3
Probability 1/8 3/8 3/8 1/8
HMM HHM MHM HMH MMM MMH MHH HHH
The mean of a random variable X is also called
expected value of X. What is the expected number
of baskets made? Do the computations...
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• Weve already discussed the mean of a density
  curve as being the balance point of the curve
  to establish this mathematically requires some
  higher level math So well think of the mean of
  a continuous r.v. in this way. For a discrete
  r.v., well compute the mean (or expected value)
  as a weighted average of the values of X, the
  weights being the corresponding probabilities.
  E.g., the mean of Hs in 4 tosses of a fair coin
  is computed as (1/16)0 (4/16)1 (6/16)2
  (4/16)3 (1/16)4 (32/16) 2.
• In either case (discrete or continuous), the
  interpretation of the mean is as the long-run
  average value of X (in a large number of
  repetitions of the experiment giving rise to X)

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• Look at Example 4.27 on page 260 a simple
  lottery (pick 3), like the old numbers gameyou
  pay 1 to play (pick a 3 digit number), and if
  your number comes up, you win 500 otherwise,
  the bookie keeps your 1. Note that in the long
  run, your winnings are
• 500(1/1000) 0(999/1000) .50
• Law of Large Numbers Essentially states that if
  you sample from a population with mean m, then
  the sample mean (x-bar) will approximate m for
  large sample sizes. Or that m is the expected
  value of many independent observations on the
  variable. CAREFULLY READ PAGES 273ff ON THE LAW
  OF LARGE NUMBERS AND ITS CONSEQUENCES! Stop
  Chapter 4 at the bottom of page 266 ("Rules for
  means"). HW Read sections 4.3 4.4. Do
  4.53-4.58, 4.61-4.63, 4.66, 4.74-4.76