Continuous Random Variables

1
Chapter 5

• Continuous Random Variables

• Continuous Random Variables
• Probability Density Function
• Expectation and Variance of Continuous Random
  Variables
• The Uniform Random Variable
• Normal Random Variable
• Exponential Random Variable
• Other Continuous Distributions
• The Distribution of Function of a Random Variables

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Objectives

• To calculate the expectation and variance of
  continuous random variables
• To focus on the uniform, normal and exponential
  random variables
• To determine the distribution of a function of a
  random variable

3
Continuous Random Variables

• Introduction
• Consider n uniform sample values in the interval
  0, 1 and a sample value is randomly selected.
• Let Xi denote the value selected, so Xi is a
  discrete random variable
• ---n increase, the probability decrease
• Now, a value is selected from 0, 1. Let X
  denote the value selected.
• ---for any x from 0, 1
• Therefore, the probability mass function is not
  suitable to describe X.

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Continuous Random Variables

• So, we use relative likelihood to describe the
  information that all the possible outcomes are
  equally likely.
• Next, consider a value from 0, 2 is randomly
  selected.
• Comparing the relative likelihood for the
  interval 0, 1 and 0, 2
• X, Y are continuous random variables. fx(x),
  fy(y) are probability density function (p.d.f.).

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Continuous Random Variables

• Definition
• A random variable, say X , is called continuous
  random variable if there exists a non-negative
  function fX, where having
  the property that for any set B of real numbers
• f is called the probability density function of
  the random variable
• The probability that X in B can be obtained by
  integrating the probability density function over
  the set B.
• Hence, all probability statements about X can be
  obtained by fX

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Continuous Random Variables

• Let a is same as b
• Therefore, the probability of a continuous
  random variable at any fixed value is equal to
  zero. Hence

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Probability Density Function

• Properties of p.d.f.

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Relations Between CDF and PDF

• -------relation between c.d.f and
  p.d.f.
• Note FX(x) is differentiable everywhere except a
  finite number of points, say x1, x1, , xn , then

9

• Ex 5.1
• The amount of running time of a computer program
  is a continuous random variable with the
  probability density function
• What is the probability that the computer run
  the program within 10 and 50 units.
• Ans
• First we calculate the value of a
• Hence, the probability that the computer run the
  program within 10 and 50 units is given by
• Therefore, the probability is 0.524

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Expectation and Variance

• Expectation of continuous random variable
• For discrete random variable X, the expectation
  is defined by
• If X is a continuous random variable having the
  probability density function f(x), then
• Hence
• If Y is a non-negative random variable, then
• If X is a continuous random variable with
  probability density function fX (x), for any
  real-valued function g

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Expectation and Variance

• Proof
• If Y is a non-negative random variable, let fY
  be the probability density function of Y.
• For a non negative function g,

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Expectation and Variance

• Variance of continuous random variable
• If X is a continuous random variable with
  expected valueµand has the probability density
  function f(x), then
• Properties
• 1
• 2

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Ex 5.2 If X has density function f with f(x) 0
when xlt0, and distribution function F, then
y
y
yx
yx
x
x
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• Ex 5.3
• Find EX and Var X when the probability
  density function is
• Ans
• The expectation The variance

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• Ex 5.4
• The probability density function of a random
  variable X, is
• If Y 2X, find EY
• Ans

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The Uniform Random variable

• Definition
• A random variable X is defined to be uniform
  random variable on the interval (a, b) if its
  probability function is given by
• Hence,
• so f(x) is a probability density function.

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• Ex 5.5
• If X is a uniform random variable in the
  interval a, b, find the expectation, EX and
  the variance, Var (X).
• Ans

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• Ex 5.6
• If X is a number randomly selected from the
  interval (0, 1), find the probability that the
  second digit in square root of X is equal to k,
• Ans
• Let Ak denote the set of numbers in (0, 1) whose
  square roots have a second digit equal to k.
• As X is uniformly distributed over (0, 1), the
  probability density function is

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• Ans
• Hence, the probability is 0.0910.002k

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• Ex 5.7
• A point is chosen at random along a line of
  length 1, dividing the line into two segments.
  What is the expected value of the ratio of
  shorter segment to longer segment
• Ans
• Let X denote the ratio chosen. The probability
  density function is
• Define the function of the ratio be g(X),

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• Ex 5.8
• An one unit length long stick is split at point
  P that is uniformly distributed over (0, 1). Find
  the expected length of the piece that contains
  the point Q, where Q is measured from the zero
  point of the stick
• Ans
• Let L(P) denote the length of the piece that
  contains the point Q
• The probability density function

0
Q
P
L(P)
0
Q
P
L(P)
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Ex 5.9 Let X be a random variable such that Pr
0 X c 1. Show that
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Ex 5.10 Let X be a random variable with
distribution function F(X). Then the random
variable Y F(X) has uniform distribution over
(0, 1). Proof
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The Normal Random variable

• Definition
• A random variable X is defined to be normal
  random variable (or say X is normally
  distributed) with parameters µ and s2 if its
  probability density function is given by
• We can also write to
  represent the relationship
• so fX (x) is a probability density function.

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The Normal Random variable

• Expectation
• Therefore, the expectation of the normal random
  variable is the parameter µ.

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The Normal Random variable

• Variance
• Integration by part
• Therefore, the variance of the normal random
  variable is s2.

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The Normal Random variable

• For
• Note
• 1. The mean and variance of a normally
  distributed random variable uniquely determine
  the probability density function.
• 2. Normal distribution is also called Gaussian
  distribution.
• Physical meaning
• µdetermines the axis of symmetry of the plot of
  fX (x).
• sdetermines the focus or the spread of fX (x)
  about this axis of symmetry.
• (the larger s, the greater the spread)

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The Normal Random variable

• Feature

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The Normal Random variable

• Approximation
• To find the probability such as, F(x) or P(a lt X
  ltb) when X is a normal random variable, we need
  use only one table which show the area under the
  standard normal curve to the left.
• as the probability density function fX
  (x) is symmetrical about the y axis

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Ex 5.11 X is N(0, 1) random variable. Then
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Ex 5.12 If X is normal random variable with µ 3
and s2 9, find (a) Pr 2lt Xlt5 (b) Pr
Xgt0 (c) Pr X-3 gt 6
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Ex 5.13 Suppose that a binary message either 0
or 1 must be transmitted by wire from A to B.
If X 2 is the value sent at A corresponding to
0 and 1 respectively. Then the value R received
at B is R X N where N is the channel noise
which is a normal random variable N(0, 1) If
R0.5, then 1 is concluded. If Rlt0.5, then 0 is
concluded. There are two types of errors Pr
error message is 1 Pr Rlt0.5
X2 Pr 2Nlt0.5 Pr N lt
-1.5 1-F(1.5) 0.0668 and Pr error
message is 0 Pr R0.5 X-2 Pr
-2N0.5 Pr N 2.5 1-
F(2.5)0.0062
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Ex 5.14 X is normally distributed with
parameters µ and s2, then Y aXß is normally
distributed with parameters aµß and a2 s2, agt0
Proof
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• Ex 5.15
• The lifetime T of a transistor is normally
  distributed with mean 200 hours and standard
  deviation 9. What is the probability that a
  particular transistor will last more than 210
  hours? What is the probability that a transistor
  which lasts 200 hours will last more than 220
  hours
• Ans.
• Although T is a non-negative random variable,
  the actual probability density function of T,
  f(t), is zero for negative t, we let the
  following p.d.f to approximate f(t)
• The probability of lasting more than 210 hours
  is
• If it lasts 200 hours, the probability of
  lasting 220 hours too is

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The DeMoivre-Laplace limit theorem If Sn denotes
the number of successes that occur when n
independent trials, each resulting in a success
with probability of p, are performed then , for
any a lt b Ex 5.16 Let X be the number of
times that a fair coin, flipped 40 times, lands
heads. Find the probability that X20. PX20
P 19.5 X 20.5
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The Exponential Random Variable

• Definition
• A random variable X is defined to be exponential
  random variable (or say X is exponentially
  distributed) with positive parameter ? if its
  probability density function is given by
• so fX(x) is a probability density function.
• The cumulative distribution function

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The Exponential Random Variable

• Expectation
• Variance

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Moment of exponential random variable
Proof
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• Ex 5.17
• A system consists of two processors and three
  peripheral units. The system is functioning as
  long as one processor and two peripherals are
  functioning. It is known that the processor
  lifetimes are exponential random variable with
  parameter a and the peripheral lifetimes are
  exponential random variables with parameter b.
  All the components fail independently. Let T be
  the lifetime of the system. Find the c.d.f. of
  the lifetime.
• Ans
• Let Xi be the lifetime of processor i, i1, 2
• Let Yj be the lifetime of peripheral j, j1, 2,
  3

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• Ans
• Let A be the event both processors fail to last
  for t
• Let B be the event at least two peripherals
  fail to last for t
• Therefore, the c.d.f. of the lifetime of the
  system is

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Other Continuous Distributions

• The Gamma Distribution
• Note

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Other Continuous Distributions

• The Weibull Distribution
• The Cauchy Distribution

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Other Continuous Distributions

• The Beta Distribution

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The Distribution of a Function of a Random
Variable

• Here, the probability density function of a
  random variable X is known. How do we determine
  the probability density function of some
  functions, say g(X) of it.
• Method 1 Express the event that in
  terms of X being in some set.
• Method 2 If g(X) is a strictly monotone(either
  increasing or decreasing) and differentiable
  function of X , use the below formula to find
  the probability density function of random
  variable Y defined by Y g(X).

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The Distribution of a Function of a Random
Variable

• Proof
• Case 1 y g(x) is a strictly increasing
  function.

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The Distribution of a Function of a Random
Variable

• Proof
• Case 2 y g(x) is a strictly decreasing
  function.

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The Distribution of a Function of a Random
Variable

• Summary
• If y g(x) is a continuous and monotonic
  function over R, then Y g (X) has the
  probability density function

since
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• Ex 5.18
• If the probability density function fX(x) of
  random variable X is given, find the probability
  density function fY(x) of .
• Ans
• The cumulative distribution function of Y

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• Ex 5.19
• If the probability density function fX(x) of
  random variable X is given, find the probability
  density function fY(x) of .
• Ans