More on Continuous Probability Distributions

1
More on Continuous Probability Distributions

• QSCI 381 Lecture 20

2
Approximating a Binomial Distribution-I

• It is possible to compute approximate
  probabilities for the binomial distribution using
  the normal distribution.
• If np ? 5 and nq ? 5, the binomial random
  variable X B(n,p) is approximately normally
  distributed
• This approximation can be quite accurate even
  though the binomial distribution is discrete and
  the normal distribution is continuous.

3
Approximating a Binomial Distribution-II

• We need an interval when defining the probability
  associated with a discrete value for x using the
  normal distribution, i.e. PB(Xx) is approximated
  byPN(x-?ltXltx?).
• We call ? the correction for continuity and set
  it to 0.5.

4
Approximating a Binomial Distribution-III
n30 p0.25
5
Approximating a Binomial Distribution-IV(Guidelin
es)

• Specify the values for n, p, and q.
• Can the normal distribution be used for
  approximation purposes, i.e. is np?5, nq?5?
• Find the mean and standard deviation for the
  approximating normal distribution.
• Apply the continuity correction.
• Find the approximate z-score and calculate the
  probability.

6
Approximating a Binomial Distribution

• 175 salmon enter a stream. If the probability of
  reaching the spawning ground is 0.3, what is the
  probability that less than 50 reach the spawning
  grounds?
• n175 p0.3 np52.5 nq122.5.
• ?52.5 ?6.06
• x 49.5 so z (49.5-52.5)/6.06-0.495
• Using NORMDIST, this value of z corresponds to
  P0.31

7
Approximating a Poisson Distribution

• It is possible to compute approximate
  probabilities for the Poisson distribution using
  the normal distribution.
• If ?gt10, the Poisson random variable X P(?) is
  approximately normally distributed

8
Approximating a Poisson Distribution(Example-I)

• Suppose the average number of salmon passing a
  counting weir is 5 per minute. What is the
  probability that in a 15 minute period more than
  85 salmon pass the weir?

9
Approximating a Poisson Distribution(Example-II)

• We first need to express the mean rate in terms
  of numbers per 15 minute interval, ?15575.
• ?gt10 so we can use the normal approximation.
• The z-score (85.5-75)/?751.212. This
  corresponds to a probability of 0.887 of 85 or
  less salmon passing in 15 minutes or 0.113 of
  more than 85 salmon passing in 15 minutes.
• Note 85 is replaced by 85.5 because we need the
  probability of 85 or fewer animals.

10
The Exponential Distribution-I

• The Poisson distribution arose when counting the
  number of events in an interval. The exponential
  distribution considers the interval of time
  between events.
• If events are occurring at random with average
  rate of ? per interval, then the probability
  density function for the length of time between
  events is

11
The Exponential Distribution-II(Example-I)

• If a herring spawns 1.5 times per week on
  average, what is the probability that it doesnt
  spawn for 2 weeks?
• Note that this is a continuous probability
  distribution so we are looking for the area under
  the curve

(wks)
12
The Exponential Distribution-II(Example-II)

• Solution
• PXgt2 1-PX?20.0498
• I used the EXCEL function
• EXPONDIST(x,?,cum)

13
The Uniform Distribution

• This is the simplest possible continuous
  distribution. It is used to deal with the
  situation where all values in some interval (a,
  b) are equally likely.
• Notation
• The probability density function for the uniform
  distribution is

14
The Uniform Distribution(Example)

• Suppose the mass of a nominally 500 kg bag of
  fish is equally likely to lie between 480 and 505
  kg. What is the probability that the bag contains
  at least the advertised mass of fish?