Discrete Probability distribution Discrete Probability distribution

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1 438//9.4.. Discrete Probability distribution.4.. Binomial P.D. The outcomes belong to either of two relevant categories. A binomial experiment requirements: o There is a fixed number of trials (n). o On each trial there are possible outcomes (success & failure). o P(X:success)=p & P(X:failure)=-p o The probabilities must remain constant for each trial. o The trials are independent..4.. Discrete Probability distribution.4.. Binomial P.D. (Examples) Flip a coin 5 times. The two outcomes in each trial are heads and trials. A drug can either cure or not cure a patient. A certain drug has probability 0.9 of curing a disease. It is administered to 00 patients.

2 438//9.4.. Discrete Probability distribution.4.. Binomial P.D. (Calculation) Binomial distribution can be determined by: otable (Appendix-Table -BiP) oformula: n! P(X = x) = x! (n x)! px ( p) n x (.0) Cumulative binomial probability: P(X x) = P(x i ) x i x (.0b) Continuous probability distribution A continuous random variable is one that can assume an uncountable number of values. This type of random variable is very different from the discrete one: o We cannot list these possible values. o The possibility of each individual value is equal to 0. we can only determine the probability of a range of values. 4

3 438//9 Probability density function Construct a relative frequency histogram. The sum of the relative frequencies (probabilities) along the whole range equals. How can the area in all the rectangles together add to? P(a<variable<b)=Area red zone Uniform distribution The values spread evenly over the full range of possibilities. f(x) = b a where a x b f(x) 6 3

4 438//9.4.. Normal distribution The normal distribution is the most important form of distributions because of its crucial role in statistical inference. f(x) = σ π e x μ σ < x < Normal distribution (facts) The normal distribution is determined by parameters. The curve is symmetric about its mean The tails of the curve are asymptotic to the horizontal axis P μ σ x μ + σ = P μ σ x μ + σ = P μ 3σ x μ + 3σ =

5 438// Standard normal distribution we will need a separate table for normal probabilities for a selected set of values of mean and standard deviation. we can reduce the number of tables needed to one by standardizing the random variable The standard normal random variable: x μ z = σ The standard normal distribution has a mean of 0 and a standard deviation of Probability and z scores P(x < X < x ) = x x σ π e x μ σ dx P(z < Z < z ) = z z π e z dz 0 5

6 438// Probability and z scores Finding probabilities when given z (table or formula) Finding z when given probabilities (table or formula) Appendix- Table -SND z Probabilities 3.9 P(z < Z < z ) = erf z erf z.4..3 Student t distribution This distribution is very common in statistical inference. The Student pdf of random variable of t is f(t) = Γ ν + νπ Γ ν + t ν ν+ 6

7 438//9 Probabilities and random variable of t Finding probabilities when given t (table or formula) Finding t when given probabilities (table or formula) Appendix- Table 3-tD v probabilities t values + P(t > t A,ν ) = A = f(t) dt t A,v Chi-squared distribution This distribution is used widely in statistical inference. The pdf of the chi-squared distribution is: f(χ ) = ν Γ ν χ ν e χ χ > 0 4 7

8 438// Chi-squared distribution v Appendix- Table 4-ChiD 00 Areas to the right of the critical value (A) The critical value χ A,ν The L-tail value= The R-tail value χ A,ν F distribution This distribution is used widely in statistical inference. The pdf of the F distribution is: f(f) = Γ ν + ν Γ ν Γ ν F > 0 ν ν ν F ν + ν ν F ν +ν, 6 8

9 438// F distribution Appendix- Table 5-FD for different A values P(F > F A,ν,ν ) = A v v 0 The critical value F A,ν,ν F A,ν,ν = F A,ν,ν 7.5 Population Characteristics Probability distribution parameters A probability distribution is in fact a model of theoretically perfect population frequency distribution. Mean and variance of frequency table applied to population: μ = x i. P(x i ) σ = x i μ. P x i = x i. P x i μ 8 9

10 438//9.5 Population Characteristics.5. Expected value The expected value of function g(x) is given by: Discrete random variable E g(x) = g(x i ) p(x i ) Continuous random variable E g(x) = g(x) f(x) dx If g(x)=x Discrete random variable E X = x i p(x i ) Continuous random variable E X = x f(x) dx 9.5 Population Characteristics.5. Expected value (conclusions) We may conclude that: E X = μ V(X) = σ = E X μ X = E X μ Then: Discrete random variable V(X) = σ = x i μ. P i (x) Continuous random variable V(X) = σ = x μ. f(x) dx 0 0

11 438//9.5 Population Characteristics.5. Expected value (extensions) E c = c E(X + c) = E(X) + c E(cX) = ce(x) Var(c) = 0 Var(X + c) = Var(X) Var(cX) = c Var(X)