Probability Density Functions

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Tutorial 2

• Probability Density Functions Normal Density

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Probability Density Function

• Unlike discrete random variables, continuous r.v.
  can take infinite no. of values.
• In this case, the probability that the r.v. takes
  a particular value is zero, i.e.
• P(X x) 0
• We cant represent the distribution of X by
  probability mass function like the discrete case.

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Probability Density Function

• We use probability density function f(x) to
  represent the distribution of a continuous r.v.
• The value of f(x) is not a probability. Instead,
  the integral of f(x) gives the required
  probability.
• Several important properties can identify a p.d.f.

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Properties of p.d.f.

• f(x) ? 0

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Ex. of p.d.f.

• Ex. (P.71 Q.2)
• Suppose you choose a real no. X from the
  interval 2,10 with a density function of the
  form f(x) Cx, where C is a constant.
• (a) Find C.

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Ex. of p.d.f.

• (b) Find P(E), where Ea,b is a subinterval of
  2,10.

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Ex. of p.d.f.

• (c) Find P(X gt 5), P(X lt 7) and P(X2 -12X 35 gt
  0) .

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Normal Density

• One commonly used distribution is the normal
  density. Its p.d.f is given by
• where -? lt X lt ? .
• mean ? , std. dev. ?

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Normal Density

• Assume X is a normal r.v. with mean ? and std.
  dev. ?.
• Let Z (X - ?)/? .
• Z is normal r.v. with mean 0 std. dev. 1
• We said Z has a standard normal distribution.
• As it is hard to obtain the prob. of a normal
  distribution from the integral of its p.d.f.,
  P(Zlta) has been calculated in a table for use.

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Normal Density

• By transforming any normal r.v. X to Z, P(X lt b)
  can be obtained from the table, i.e.
• P(X lt b) P((X - ?)/? lt (b - ?)/?)
• P(Z lt (b - ?)/?)
• P(Z lt a)

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Normal Density

• Ex.(P.222 Q.25(d))
• Let X be a r.v. normally distributed with ?
  70, ? 10. Estimate P(60 lt X lt 80).