Understanding Probability Distributions: Discrete vs. Continuous Options
Delve into the world of probability distributions with distinctions between discrete and continuous distributions, explained through examples like dice rolling and child gender probability. Explore the significance of the normal distribution in medical data analysis.
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Probability (P) Discrete probability distribution Continuous probability distribution Assist. Prof. Dr. Humam G. Zubeer Family &Community Med. Depart. College of Medicine-University of Mosul October 2023
Discrete probability distribution: Suppose an event (E) can happened in (r) ways out of total of (n) possible equally likely ways. e.g: S = {1,2,3,4,5,6} (sample space of a die) Then, The prob. of occurrence of the event called it s success . Is defined by (p) where: r E p = = ) Pr( r p = n n Example of coin: 1 ) ( = =n n 2 1 n r r = 5 . 0 = = = = P H ( ) 1 ( ) 5 . 0 P T P H P of failure (q) 2 2
Where : p : Probability of an event (success). n : Sample space (success & failure events). (all events may be occur). r : Success events only. n-r : Failure events only. q - 1 = p = + q 1 p P(E) 0 (+ve) 0 p 1 If p = 1 called the Certain prob. If p = 0 called the Impossible prob.
Example: A couple planed to have 5 children. What is the possibility that 3 of them are boys? Answer: n= 5, x= 3, p=1/2=q ( n r r n r = = Pr( ) X r p q ( 5 = = ) Pr( ) 3 ( 3 ) 3 ( X 5 3 1 1 2 2 ! 5 1 1 5 4 3 2 1 1 = = = . 0 31 ( 1 ) 1 5 ( ! 3 3 )! 8 4 3 2 2 32
Probability Distribution of a Continuous Variable THE NORMAL DISTRIBUTION or GAUSSIAN DISTRIBUTION
Suppose that we increase the no. of children to 50 000 and decrease the width of interval to 0.01 lb.
The term normal curve, in fact, refers not to one curve but to a family of curves, each characterized by a mean and a variance 2. In the special case where = 0 and 2= 1, we have the standard normal curve. The curve is bell-shaped with the tails dipping down to the baseline. In theory, the tails get closer and closer to the baseline but never touch it, proceeding to infinity in either direction. In practice, we ignore that and work within practical limits.
Normal Distribution or Gaussian Distribution. Why used it? The normal distribution is important because the distribution of many medical population approximate the normal in shape. measurement in =~ = x x m x e.g: Serum pressure, glucose level, age, height, weight, . etc. uric acid level, cholesterol level, blood
2= 5 2= 10 Family of normal curves: (a) two normal distributions with the same mean but different variances; (b) two normal distributions with the same variance but differentmeans.
Sometimes the bell is tall and narrow, and sometimes it is more flattened, but whatever the dispersion of the data, approximately 95% of measurements lie within two standard deviations either side of the mean.
Characteristics of the normal distribution: 1- The general form of the normal distribution is: ~ N X 2 ( , ) Where: 2- The normal distribution is determined by two quantities the mean ( ) & variance ( 2). are ( two parameters ) 2 ( , )
Shape of Normal Distribution + 68% 2 + 2 95% + 3 3 99%
3- The normal distribution is Uni-Model (only one peak). 4- The ( Mean = Median = Mode ) 5- Bell shaped, symmetrical about the mean ( ) and the mean ( ) divided the whole area into two parts (50%) below it & (50%) above it. 6- The intervals spanning the mean ( ) by one , 2 , 3 on the both sides of mean is respectively: (68.27%), the total area. (95.45%), (99.73%) of
Lower and upper limits of the data = = U, L U , L Mean SD 68% = Mean = 95% U, L 2 U, L 2SD 3 = Mean = U, L U , L 3SD 99 %
