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Probability distributions

Probability distributions. Example. Variable G denotes the population in which a mouse belongs G=1 : mouse belongs to population 1 G=2 : mouse belongs to population 1 Probabilities for the two alternatives define a probability distribution of G P(G=1)=0.833 P(G=2)=0.167

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Probability distributions

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  1. Probability distributions

  2. Example • Variable G denotes the population in which a mouse belongs • G=1 : mouse belongs to population 1 • G=2 : mouse belongs to population 1 • Probabilities for the two alternatives define a probability distribution of G • P(G=1)=0.833 • P(G=2)=0.167 …if the sum of the probabilities is equal to 1: P(G=1)+P(G=2)=0.833+0.167=1

  3. Probability distribution as a function • Probability distribution may be defined by a set of probabilities for the alternative values of a variable • Or by a function which assigns the probabilities to alternatives • This is especially useful when there are many alternatives • The function usually has one or more parameters, which control how the probability is distributed to different values

  4. Example : Binomial distribution • x :Number of heads in 10 tosses of a coin • Parameter N: number of tosses • Parameter p: probability of heads in each trial x | N,p ~ Bin(N,p) P(x=k |N,p) ={ N!/(k!(N-k!)) } pk(1-p)N-k

  5. Binomial distribution for the number of heads in 10 tosses of a fair coin

  6. Continuous variables? • Infinite number of possible values between any two possible values. • ->probability of any particular value = 0 • There is probability density for each value: the “height” of probability mass at that point • There is probability between two points, found by integration • Practical calculations: • establish a dense grid of values at which to evaluate the probability density • Normalise the density by the sum of the grid: approximation of the amount of probability around each grid point

  7. Example: Normal distribution • Possible values: all real numbers • Parameter  : Mean of the probability mass, center of gravity • Parameter 2 : variance of the probability mass, controls the spread of the probability Probability density of x p(x=k| , 2 )=((22) -1/2 )exp{(k- )2 / 22 }

  8. Normal distribution

  9. Describing the probability distribution • Mean • Variance • Standard deviation • Median and other percentiles • Mode • Coefficient of variation

  10. Cumulative distribution

  11. Exercise 3 • Make a graph showing the probability density of a Normal distribution with mean = 100 and standard deviation of 10. Evaluate the density at values 50,55,60,65,…,150 • Using the grid approximation, calculate the following statistics of the distribution • Mean • Variance • Standard deviation • Coefficient of variation

  12. Exercise 3 continues • By using the grid approximation, calculate the cumulative distribution of the previously defined normal distribution • Use the graph to determine the following statistics • Median • 5% percentile • 95% precentile

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