Continuous Probability Distributions

1
Continuous Probability Distributions

• Uniform Probability Distribution
• Normal Probability Distribution
• Exponential Probability Distribution
• Other continuous probability distributions

2
Continuous Probability Distributions

• A continuous random variable can assume any value
  in an interval on the real line or in a
  collection of intervals.
• It is not possible to talk about the probability
  of the random variable assuming a particular
  value.
• Instead, we talk about the probability of the
  random variable assuming a value within a given
  interval.
• The probability of the random variable assuming a
  value within some given interval from x1 to x2 is
  defined to be the area under the graph of the
  probability density function between x1 and x2.

3
Uniform Probability Distribution

• A random variable is uniformly distributed
  whenever the probability is proportional to the
  intervals length.
• Uniform Probability Density Function
• f(x) 1/(b - a) for a lt x lt b
• 0 elsewhere
• where
• a smallest value the variable can assume
• b largest value the variable can assume

4
Uniform Probability Distribution

• Expected Value of x
• E(x) (a b)/2
• Variance of x
• Var(x) (b - a)2/12
• where
• a smallest value the variable can assume
• b largest value the variable can assume

5
Graph of the Normal Probability Density Function
6
Characteristics of the Normal Probability
Distribution

• The shape of the normal curve is often
  illustrated as a bell-shaped curve.
• Two parameters, m (mean) and s (standard
  deviation), determine the location and shape of
  the distribution.
• The highest point on the normal curve is at the
  mean, which is also the median and mode.
• The mean can be any numerical value negative,
  zero, or positive.
• The normal curve is symmetric.
• The standard deviation determines the width of
  the curve larger values result in wider, flatter
  curves.
• The total area under the curve is 1 (.5 to the
  left of the mean and .5 to the right).
• Probabilities for the normal random variable are
  given by areas under the curve.

7
Normal Probability Density Function

• where
• ? mean
• ? standard deviation
• ? 3.14159
• e 2.71828

8
Standard Normal Probability Distribution

• A random variable that has a normal distribution
  with a mean of zero and a standard deviation of
  one is said to have a standard normal probability
  distribution.
• The letter z is commonly used to designate this
  normal random variable.
• Converting to the Standard Normal Distribution
• We can think of z as a measure of the number of
  standard deviations x is from ?.

9
Exponential Probability Density Function

• where µ mean e 2.71828
• xgt0
• The exponential distribution is commonly used to
  measure time between events occurring.

10
The Gamma Distribution

• The Gamma distribution is an extension to the
  exponential distribution
• where xgt0, agt0, ßgt0
• ?(a)(a-1)! for a1,2,..
• The Chi-squared distribution, ? a2, is closely
  related to the gamma distribution

11
Student t distribution

• The student t distribution is closely related to
  the normal and gamma distributions and plays and
  important role in certain statistical testing
  procedures.

12

• Sampling and Sampling Distributions
• Simple Random Sampling
• Point Estimation
• Introduction to Sampling Distributions
• Sampling Distribution of
• Sampling Distribution of p
• Interval Estimation
• Interval estimation of a population mean large
  and small sample cases
• Determining the sample size
• Interval estimation of a population proportion
• Hypothesis Testing
• Tests about a population mean large and small
  sample cases
• Tests about a population proportion

13
Statistical Inference

• The purpose of statistical inference is to obtain
  information about a population from information
  contained in a sample.
• A population is the set of all the elements of
  interest.
• A sample is a subset of the population.
• The sample results provide only estimates of the
  values of the population characteristics.
• A parameter is a numerical characteristic of a
  population.
• With proper sampling methods, the sample results
  will provide good estimates of the population
  characteristics

14
Point Estimation

• In point estimation we use the data from the
  sample to compute a value of a sample statistic
  that serves as an estimate of a population
  parameter.
• We refer to as the point estimator of the
  population mean ?.
• s is the point estimator of the population
  standard deviation ?.
• p is the point estimator of the population
  proportion ?.

15
Sampling Distribution of p

• The sampling distribution of p is the probability
  distribution of all possible values of the sample
  proportion
• Expected Value of p
• E( p ) ?
• where ? the population proportion
• Standard Deviation of p
• sp is referred to as the standard error of the
  proportion.

16
Sampling Distribution of p

• The sampling distribution of p can be
  approximated by a normal distribution whenever
  the sample size is large.
• A sample can be considered large when both np 5
  and n(1-p) 5.

17
Interval Estimation

• A point estimate only gives us a single estimate
  for population parameter and does not take into
  account the variability in the data or the sample
  size.
• The standard error of the sampling distribution
  of x is a measure of the reliability or precision
  of x as an estimate of µ.
• The standard error can be used to construct a
  confidence interval for the population mean, µ.
• Confidence - the level of confidence that the
  interval will contain µ.
• The confidence level is normally set at 95.

18
95 Confidence Interval for a Population Mean,
µLarge Sample Case (n gt 30)
With s unknown where x is the sample
mean s is the sample standard
deviation n is the sample size
19
95 Confidence Interval for a Population Mean,
µSmall Sample Case (n lt 30)

• Population is Not Normally Distributed
• The only option is to increase the sample size to
  n gt 30 and use the large sample interval
  estimation
• procedures.
• Population is Normally Distributed and ? is
  Unknown
• The appropriate interval estimate is based on the
  t distribution.

20
t Distribution

• The t distribution is a family of similar
  probability distributions.
• A specific t distribution depends on a parameter
  known as the degrees of freedom.
• (e.g., for a problem with 20 elements, degrees of
  freedomdf20-119)
• As the number of degrees of freedom increases,
  the difference between the t distribution and
  the standard normal probability distribution
  becomes smaller and smaller.
• A t distribution with more degrees of freedom
  has less dispersion.
• The mean of the t distribution is zero.

21
95 Confidence Interval for a Population Mean,
µSmall Sample Case (n lt 30)

• where x is the sample mean
• s is the sample standard
• deviation
• n is the sample size

22
Hypothesis Testing

• Hypothesis testing can be used to determine
  whether a statement about the value of a
  population parameter should or should not be
  rejected.
• The null hypothesis, denoted by H0 , is a
  tentative assumption about a population
  parameter.
• The alternative hypothesis, denoted by H1, is the
  opposite of what is stated in the null hypothesis
  and is often referred to as the hypothesis of
  interest.

23
Null and Alternative Hypotheses about a
Population Mean

• The equality part of the hypotheses always
  appears in the null hypothesis.
• A hypothesis test about the value of a population
  mean ?? usually takes the following form (where
  ?0 is the hypothesised value of the population
  mean).
• H0 ? ?0
• H1 ? ? ?0

24
The Steps of Hypothesis Testing

• Determine the appropriate hypotheses.
• Select the test statistic for deciding whether or
  not to reject the null hypothesis.
• Collect the sample data
• Use a statistical package to compute the test
  statistic and the p-value
• If p-valuelt0.05 then reject H0
• Make sensible conclusions based on the decision
  to reject H0 or not.

25
Tests about a Population Mean Large-Sample Case
(n gt 30)

• Test Statistic
• Rejection rule
• Reject H0 if Zgt1.96 or Zlt-1.96

26
Tests about a Population MeanSmall-Sample Case
(n lt 30)

• Assuming that the population is normally
  distributed
• Test Statistic
• Rejection rule
• Reject H0 if TgttQ2.5,n-1 or Tlt-tQ2.5,n-1

27
The Use of p-Values

• The p value is the probability of obtaining a
  sample result that is at least as unlikely as
  what is observed.
• The p value can be used to make the decision in a
  hypothesis test
• Reject H0 if the p value lt 0.05

28
Null and Alternative Hypotheses for a Population
Proportion

• The equality part of the hypotheses always
  appears in the null hypothesis.
• In general, a hypothesis test about the value of
  a population proportion p usually takes the
  following form (where p0 is the hypothesized
  value of the population proportion).
• H0 p p 0
• H1 p ? p 0

29
Example

• The following nucleotide distribution was
  observed
• Question 1 Compute estimates of the nucleotide
  probabilities
• P(A) 2000/8100 0.247
• P(C) 2100/8100 0.259
• P(G) 1500/8100 0.185
• P(T) 2500/8100 0.309
• Question 2 Test the null hypothesis that the
  nucleotide probabilities are equal, that is
• H0 p1 p2 p3 p4 ¼
• using a goodness of fit test based on the
  chi-square distribution

30

• Given that
• H0 p1 p2 p3 p4 ¼,
• expected counts for A, C, G, and T are
  8100/42025
• For Chi-square Test, the following expression is
  used to find a critical value
• eexpected value (e.g., mean value)
• o actual value
• X2 (2000-2025) 2/2025 (2100-2025) 2/2025
  (1500-2025) 2/2025 (2500-2025)2/2025
• X2 250.62
• As we have 4 elements,
• Degrees of freedom (df) 4 - 1 3
• from the Chi-Square Distributions table (see next
  page), the critical value for 95 confidence
  interval (or 5 significance level alpha) is
  found to be 7.8147 (Note that if a confidence
  level is not specified in the question, you may
  consider this as 95)
• Conclusion Reject H0 at 5 significance level as
  X2 is higher than the critical value. There is
  evidence that the probabilities are not all
  equal. There are more A and T than expected. It
  can be seen from the Question 1

31
(No Transcript)