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Probability and Statistics Part 2: Statistics. Lecture 2. What is the probability of the weight at birth of any baby born being in 2440 g class?. Frequency distribution. p=0.0677. Birthweight < 3000 g How many of them?. Frequency distribution. 28.57%. Frequency distribution.
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Probability and Statistics Part 2: Statistics Lecture 2
What is the probability of the weight at birth of any baby born being in 2440 g class? Frequency distribution p=0.0677
Birthweight < 3000 g How many of them? Frequency distribution 28.57%
Frequency distribution We sampled from unknown population of babies and found that the very first individual sampled had birthweight equal to 6000 g. Is this the same population?
P(Birthweight=6000) → 0 Frequency distribution
Frequency distribution We would reject any hypothesis that the unknown population was the same as the sampled. We would conclude, that the most probably, the unknown population we sampled from is different in mean and possibly in variance.
Frequency distribution We have used the empirical frequency distribution to make certain predictions or make judgments and decisions. In many cases we will make such predictions not from empirical distributions, but on the basis of theoretical considerations. We may feel, that data should be distributed in a certain way. If our observed data do not sufficiently conform to the values expected on the basis of these assumptions – we will have serious doubts about our assumptions.
Probability distribution The assumptions being tested generally lead to a theoretical frequency distribution, known also as a probability distribution.
Discrete probability distributions • The binomial distribution • The geometric distribution • The hypergeometric distribution • The Poisson distribution
The binomial distribution • Suppose that n independent experiments, or trials, are performed, where n is a fixed numer, and that each experiment results in „success” with probability p and a „failure” with probability q=1-p. • The total number of successes, X, is a binomial random variable with parameters n and p.
The binomial distribution The probability, the X=k, or p(k), can be found in the following way: • Any particular sequence of k successes occurs with probability • The total number of such sequences is
The binomial distribution The expected value is equal to: and variance can be obtained from:
The binomial distribution N=10, p=0.1 N=10, p=0.5
Example • Tay-Sachs disease is a rare but fatal disease of genetic origin occuring chiefly in infants and children. If a couple are both carriers of Tay-Sachs disease, a child of theirs has probability 0.25 of being born with the disease. If such a couple has four children, what is the frequency function for the number of children that will have the disease?
The binomial distribution 0.422 0.316 0.211 0.047 0.004
The geometric distribution • The geometric distribution is also constructed from independent Bernoulli trials, but from an infinite sequence. On each trial, a success occurs with probability p, and X is the total number of trials up to and including the first success. In order that X=k, there must be k-1 failures followed by a success.
The geometric distribution The expected value is equal to: and variance can be obtained from:
The hypergeometric distribution • Suppose that an urn contains n balls, of which r are black and n-r are white. • Let X denote the number of black balls drawn when taking m balls without replacement. So
The hypergeometric distribution The expected value is equal to: and variance can be obtained from:
Example Suppose there are 100 floppy disks and we know that 20 of them are defective. What is probability of drawing zero to two defective floppies if you select 10 at random? n=100 r=20 m=10
The Poisson distribution • The Poisson distribution can be derived as the limit of a binomial distribution as the number of trials n approaches to infinity and the probability of success on each trial p approaches zero in such a way that np=λ.
The Poisson distribution The expected value is equal to: and variance can be obtained from:
Example • Two dice are rolled 100 times, and the number of double sixes X is counted. The distribution of X is binomial with n=100 and p=1/36=0.0278. Since n is large and p is small, we can approximate the binomial probabilities by Poisson probabilities with λ=np=2.78
Another example • Suppose that an office receives telephone calls as a Poisson process with λ=0.5 per minute. The number of calls in a 5-min interval follows a Poisson distribution with parameter ω=5λ=2.5. Thus, the probability of no calls in 5-min interval is
Continuous density functions For the continuous random variable, the role of the frequency function is taken by a density function f(x) which has the properties: and
Continuous density functions • The uniform density • The exponential density • The normal distribution
The uniform density function • A distribution which has constant probability is called uniform distribution. • The probability density function (pdf):
The exponential density function • The exponential random variable can be used to describe the life time of a machine, industrial product and Human being. Also, it can be used to describe the waiting time of a customer for some service. • The probability density function (pdf):
The exponential density function The expected value is equal to: and variance can be obtained from:
The exponential density function and is called the survival function.
Example Let X represents the life time of a washing machine. Suppose the average lifetime forthis type of washing machine is 15 years. What is the probability that this washing machine can be used for less than 6 years? Also, what is the probability that this washing machine can be used for more than 18 years?
0.0667 0.0447 Example P(X≤6) ≈ 0.0447·6+(0.0667-0.0447)·6/2 = 0.3342
Example Thus, for this washing machine, it is about 30% chance that it can be used for quite a long time or a short time.
Poisson and exponential ... Let Y be a Poisson random variable representing the number of occurrences in theunit time interval with the probability distribution whereμ is the mean number of occurrences in this time interval. Then, if X represents the time of one occurrence, X has the exponential density function with mean
Example • The average number of car accidents on the highway in two days is 8. What is the probability of no accident for more than 3 days?
Example • The average number of car accidents on the highway in one day is 4. Thus, the mean time of one occurrence is 0.25 (day). • Let Y be the Poisson random variable with mean 4 representing the number of car accidents in one day, while X be the exponential random variable with mean representing the time of one accident occurrence.
Example P(no accident for more than 3 days) = P(the time to the first occurence larger than 3)
The normal distribution • The normal distribution plays a central role in probability and statistics. This distribution is also called the Gaussian distribution after Carl Friedrich Gauss, who proposed it as a model for measurement errors (in 1809). The normal distribution has been used as a model for such diverse phenomena as a person’s height, the distribution of IQ scores, and the velocity of gas molecules.
The normal distribution • The density function of the normal distribution depends on two parameters, μ called mean and σ named standard deviation of the normal density (where -∞< μ< ∞ and σ > 0):
The normal distribution μ=0 μ=4
The normal distribution σ=1 σ=2 σ=3
The normal distribution • The curve is symetrical around the mean. Therefore the mean, median, and mode of the normal are the same. • The following percentages of individuals in a normal distirbution lie within the indicated limits: μ±σ contains 68.72 % of individuals μ± 2σ contains 95.45 % of individuals μ± 3σ constains 99.73% of individuals