Distributions Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics

Distributions Three distributions are backbone of medical statistics Binomial Distribution Poisson Distributions Normal Distributions Each distribution has a formula known as probability distribution function, this gives the probability of an event, known as parameters, which identify the particular distribution, and from which various characteristics of the distribution such as its mean, standard deviation, can be calculated

Binomial distribution Binomial distribution is measured on a binary nominal scale Success / Failure, where number of successes is X, N Identical trial. The total number of successes is discrete ratio scale. Trial are independent. The probability of success the same for each trial.

The characteristics of binomial distribution is µ = n π The Standard Deviation is

N= number of trials (cases) X= number of successes π = probability of success in each trial Example if you flipped a coin twice, calculate the four outcomes probabilities

Number of HeadsProbability

Example2 The probability of getting ill per a day is P=0.10, and there are twp persons N=2, what is the probability that neither of them is ill, means X=0.

The probability of having zero successes from these two is 0.81 What is the probability that at least one will be ill? Means X>0 XProbability

P(X>0)=P(X=1)+P(X=2)= =0.19 Or we can calculate it by taking 1- P(x≤0).

Poisson Distribution Poisson distribution is used to describe discrete quantitative data such as counts that occur independently and randomly in time or space at some average rate. The Poisson distribution has the following properties: The mean of the distribution is equal to μ. The variance is also equal to μ.variance

Poisson Distribution (Conti.) A Poisson experiment is a statistical experiment that has the following properties:statistical experiment The experiment results in outcomes that can be classified as successes or failures. The average number of successes ( μ ) that occurs in a specified region is known. The probability that a success will occur is proportional to the size of the region. The probability that a success will occur in an extremely small region is virtually zero.

e: A constant equal to approximately (Actually, e is the base of the natural logarithm system.) μ : The mean number of successes that occur in a specified region. x: The actual number of successes that occur in a specified region. P(x; μ ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.

Example The average number of Emergency cases in a hospital is 2 cases per day. What is the probability that exactly 3 emergency cases will be received tomorrow? Solution: This is a Poisson experiment in which we know the following: μ = 2; since 2 emergency cases are received per day, on average. x = 3; since we want to find the likelihood that 3 emergency cases will be received tomorrow. e = ; since e is a constant equal to approximately We plug these values into the Poisson formula as follows: P(x; μ ) = (e - μ ) ( μ x ) / x! P(3; 2) = ( ) (2 3 ) / 3! P(3; 2) = ( ) (8) / 6 P(3; 2) = 0.180

Normal Distribution The normal distribution refers to a family of continuous probability distributions described by the normal equation.continuous probability distributions where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately , and e is approximately

The normal distribution is a continuous probability distribution. This has several implications for probability. The total area under the normal curve is equal to 1. The probability that a normal random variable X equals any particular value is 0. The probability that X is greater than a equals the area under the normal curve bounded by a and plus infinity (as indicated by the non-shaded area in the figure below). The probability that X is less than a equals the area under the normal curve bounded by a and minus infinity (as indicated by the shaded area in the figure below).

Additionally, every normal curve (regardless of its mean or standard deviation) conforms to the following "rule". About 68% of the area under the curve falls within 1 standard deviation of the mean. About 95% of the area under the curve falls within 2 standard deviations of the mean. About 99.7% of the area under the curve falls within 3 standard deviations of the mean.

Example 1 An average expiry of medicine manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that medicine life is normally distributed, what is the probability that an Acme medicine will last at most 365 days? Solution: Given a mean score of 300 days and a standard deviation of 50 days, we want to find the probability that medicine life is less than or equal to 365 days. Thus, we know the following: The value of the normal random variable is 365 days. The mean is equal to 300 days. The standard deviation is equal to 50 days.

We substitute these values into the Normal Distribution formula and compute the probability. The answer is: P( X < 365) = Hence, there is a 90% chance that a medicine will last for within 365 days.