CONTINUOUS RANDOM VARIABLES

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CONTINUOUS RANDOM VARIABLES

• These are used to define probability models for
  continuous scale measurements, e.g. distance,
  weight, time
• For a large data set we summarise the
  distribution using a relative frequency histogram
  

the relative frequency of observations between a
and b is proportional to the areas of the
rectangles above a,b.
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Relative Frequency Histogram

• As sample size increases

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Approximation by normal distribution

• Histogram of 1000 obs. Normal curve overlaid

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Using frequency curves

• Frequency curves are drawn so that the area under
  the curve is one. So, the area to the left of
  any value on the x-axis is merely the proportion
  of the population which falls below that value.
• What proportion of the ____ are less than 39?
  The distribution reveals its 30

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Frequencies for the normal distribution
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Probability Density Function (Freq. Curve)

• For a continuous random variable X, we describe
  the probability distribution by some function
  f(x) e.g.

such that (i) f(x) gt 0 for all x (ii) area
under the curve between a and b is which is P(
a lt X lt b) (iii) Total area under curve 1.
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Probability density function and c.d.f

• f(x) is called the probability density function
  (p.d.f.) of X.
• For a continuous random variable the probability
  of it taking a particular value exactly, e.g. X
  length of a bolt 1.999965722 cms, is zero. That
  is PX x 0
• Instead for continuous random variables
  probabilities are associated with a range of
  values.e.g. 1.95 ? X ? 2.00 cms.
• The cumulative distribution function (c.d.f.)
  F(x) is defined as the probability upto x, i.e.
  F(x) P(X ltx)

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Example - Uniform Continuous Distribution
X can take any real value between a and b with
probability uniform over this interval.

• Total area 1 length x height
• Thus the probability density function is

Generating 10 uniform random variables in
S-plus unifrv10_runif(n10, mina, maxb)
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Uniform Continuous Distribution

• For any values c and d between a and b

C.d.f. F(x)
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Expectation and variance
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Normal distributions

• Normal distributions are one type of continuous
  p.d.f.
• If X has the Normal distribution with mean µ and
  variance ??2, this is denoted by XN(µ,?2)
  (Splus uses s.d. instead of var)
• Z N(µ0,?21) is called the standard normal
  distribution
• Since normal probabilities are hard to compute,
  tables were made for the standard normal
  distribution only
• Most textbooks give areas under the curve of the
  N(0,1) p.d.f

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Calculating standard normal probabilities
Find the probability of getting a value of Z
greater than 1.05
P(Zgt1.05) 1 - P(Zlt1.05) look up P(Zlt1.05) in
tables P(Zlt1.05) 0.8531 P(Zgt1.05)
Find the prob of Z between -1.05 and 1.05
P(-1.05ltZlt1.05) P(Zlt1.05) - P(Zlt-1.05) 0.8531
- P(Zgt1.05)
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• In order to obtain probabilities for other Normal
  distributions (i.e. areas under the curve), it
  is necessary to express any value of X in terms
  of the number of standard deviation units it is
  away from µ.

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Example of normal distribution

• A filling machine is used to fill soft drink
  bottles. The bottles are supposed to contain 300
  mls. In fact the quantities vary according to the
  Normal distribution with expected value of µ
  302 ml and standard deviation s 3ml.
  What is the probability that an individual bottle
  contains less than 295 mls?
• Let the r.v. X denote the quantity in an
  individual bottle. We are told X N(302, 32),
  and we want PrX lt 295.
• If X 295 then Z (295 - 302)/3
  -2.33
• so P(X lt 295) P(Z lt -2.33) 1 - P(Z lt
  2.33)
• 1 - .990 0.01
• i.e. about 1 bottle in 100 would have less than
  295 ml.

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1 and 2 sigma bands of Normal distribution
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Normal probabilities from R

• e.g. If X N(5,9)
• (i) find P(X lt 7)
• (ii) find k such that P(X lt k) 0.05
• p7_pnorm(q7, mean5, sd3) (0.7475)
• q0.05_qnorm(p0.05, mean5, sd3) (0.0654)
  
• Check using tables

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Exponential distribution

• T time to first arrival
• P(T gt t) P(N(t) 0) (lt)0 e-lt / 0! e-lt
• c.d.f of T FT(t) P(Tlt t) 1 - e-lt
• p.d.f. of T fT(t) d/dt FT(t) le-lt

E(T) 1/l V(T) 1/l2
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Residual time distribution
T
R
t
P(Rgtr) P(Tgtrt Tgtt) P(Tgtrt)/P(Tgtt) e-?(rt)
/e-?t e-?r P (T gt r)
time up to next arrival is independent of when
the previous arrival occurred

• Distr of additional lifetime is same as the
  original distr.
• Memoryless property of exponential
• Other distributions such as gamma, Weibull etc.
  are not memoryless (exponential is the only one).

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Quantiles

• Cumulative distribution function F(x)
• F(x) P( X lt x)
• e.g. Z N(0,1) P(Z lt 1.96) ?
• need to look at the c.d.f. curve to answer this
  question
• F(1.96) 0.975 ? P(Z lt 1.96) 0.975
• i.e. 1.96 is the 97.5 th percentile of N(0,1).
  1.96 F-1(0.975)
• What is the 50th percentile of N(0,1) ?
• What is the 60th percentile of N(0,1) ?
• Ans Look at quantile plot.
• Quantile plot x-axis - cumulative probability
  (0,1) y-axis F-1
• Quantile plot is inverse of c.d.f. plot.

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Normal quantiles
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Q-Q plot of Chisq distribution (vs. normal)