Continuous Random Variables. What continuous random variables are and how to use them. I can give a definition of a continuous random variable.

Transcription

1 Continuous Random Variables Today we are learning... What continuous random variables are and how to use them. I will know if I have been successful if... I can give a definition of a continuous random variable. Define the sample space of a continuous random variable. I have considered the probability density function of a continuous random variable. Continuous Random Variable A continuous random variable can take any value over a certain range. Examples include; height, age or temperature.

2 Discrete vs Continuous Random Variable Discrete R.V. Continous R.V. Range Space {1, 2, 3} {x :100 < x < 300, x R} Probability Statement P(X = 1) = P(X < 3) = P(X < 200) = P(X = 250) = 0 Continuous Random Variable P(X = x) = 0 and consequently... P(X < x) = P(X < x)

3 Continuous Random Variables For continuous random variables, probabilities can be determined by first making an assumption about the nature of the population concerned. We assume that the population can be described by a curve having a particular shape. There are 4 such shapes we consider. Symmetrical Positively Skewed Decreasing Uniform These curves, probability density functions (pdf), show how the total probability is spread across the range of values the continuous random variable can take.

4 Choosing the Correct Curve We choose the curve by comparing its shape to a histogram or making theoretical assumptions. Bus Example - Page 52 Note: f(t) represents the probability density function, pdf. F(t) represents the cumulative distribution function, cdf. The total area under the pdf should equal 1.

5 Business Example - Page 53 Summary of results are shown on Page 54 from this section. Ex 2.5A and 2.5B The Normal Distribution Today we are learning... What the normal distribution is and how we can calculate probabilities using it. I will know if I have been successful if... I know the formula for the normal distribution. I know the conditions under which it can be used. I can use different methods to help me calculate probabilities.

6 The Normal Distribution A continuous random variable X is said to have a Normal distribution if it has the probability density function... We can then write... = E(X) The Normal Distribution = V(X) The normal distribution gives a 'bell shaped' curve, where the probability density is greatest at the mean.

7 The Normal Distribution As shown in the textbook, page 58, we have many different types of Normal distribution, with different variances and means. To calculate probabilities, as we have seen for continuous random variables we integrate to find the area beneath the probability density function. Unfortunately, due to the complexity of the formula for the normal distribution we are unable to do this, without other area of mathematics. Instead this has been done for us and given in the tables. The Standard Normal Distribution Z ~ N(0, 1) We denoted the standard normal distribution with Z instead of X. It has a mean of 0 and a standard deviation of 1. The cumulative density function is denoted with in the tables. (z) and is shown

8 The Standard Normal Distribution So how would we calculate the probability of a N(2, 9) distribution, if we only have the table values for N(0, 1)? We can 'transform' the distribution using the following formula. (Formula given on Page 58) The Standard Normal Distribution

9 Combining Normal Random Variables Today we are learning... How to combine random normal variables and calculate associated probabilities. I will know if I have been successful if... I can recall the laws of expectation and variance. I can apply this to the Normal Distribution. I can calculate associated probabilities. The Laws of Expectation and Variance We have previously seen: E(X + Y) = E(X + Y) E(aX + b) = ae(x) + b Var(X + Y) = Var(X) + Var(Y)* 2 Var(aX + b) = a Var(X) *Provided X and Y are independent. We get a similar result when combing normal random variables.

10 Combining Normal Random Variables If X and Y are independent such that; Then it can be shown that; X + Y In other words the sum of two normal distributed random variables is also normally distributed. Example X ~ N(10, 5) and Y ~ N(9, 4) are independent random variables. Calculate a) P(X + Y < 17) b) P(X > Y) A similar example is given on Page 61 if you wish to read it.

11 Page 62 Approximating the Binomial with the Normal Previously we have seen that we can approximate the binomial with the Poisson distribution under the conditions that n is large (n > 20) and p is small (p < 0.05). However we can't use this when p is very large? Instead we can approximate it with a Normal distribution instead.

12 Approximating the Binomial with the Normal Today we are learning... How to approximate the Binomial Distribution with the Normal. I will know if I have been successful if... I can state the conditions which must be met to do this. I understand why we might do this. I can calculate associated probabilities. Normal Distributions in TI 84 normalcdf(lowerbound, upperbound, mean, standard deviation) If we want a lower bound of negative infinity we type -E99

13 Approximating the Binomial with the Normal If X ~ Bin(n, p) where n is large and p is close to 0.5 then approximately... X ~ N(np, npq). Rule of Thumb Use Normal when np and nq are greater than 5. Use Poisson when n is large and p is small. Otherwise we have to use the binomial formula, regardless of how complicated it gets! Example X ~ B(20, 0.5) Find by approximating the binomial with the normal P(X = 8)

14 Continuity Correction The Binomial is discrete while the Normal is a model for continuous data. So we can not evaluate P(X = x) instead we have to consider the probability over the interval x < Y < x This is what is known as a continuity correction. If X ~ B(n, p) and Y ~ N(np, npq) then P(X < x) = P(Y < x - 0.5) P(X < x) = P(Y < x + 0.5) P(X > x) = P(Y > x - 0.5) P(X > x) = P(Y > x + 0.5) Examples Suppose X ~ Bin(18, 0.45) Calculate these probabilities using an approximation. a) P(X < 9) b) P(X < 13) c) P(X > 14)

15 Approximating the Poisson with the Normal Today we are learning... How to approximate the Poisson Distribution with the Normal. I will know if I have been successful if... I can state the conditions which must be met to do this. I understand why we might do this. I can calculate associated probabilities. Approximating the Poisson with the Normal If X ~ Po( ) then we can approximate a Poisson with X ~ N(, ) Conditions: This is a suitable approximation when > 10.

16 Continuity Correction The Poisson is discrete while the Normal is a model for continuous data. So we can not evaluate P(X = x) instead we have to consider the probability over the interval x < Y < x This is what is known as a continuity correction. P(X < x) = P(Y < x - 0.5) P(X < x) = P(Y < x + 0.5) P(X > x) = P(Y > x - 0.5) P(X > x) = P(Y > x + 0.5) Example Suppose cars arrive at a car park at a rate of 50 per hour. Calculate the probability that in the next hour between 54 and 62 cars will arrive. i) Using the Poisson distribution. ii) Using a suitable approximation. Answer (i) Answer (ii) Reference: wiki.stat.ucla.edu/socr/

17 Example 2 The annual number of earthquakes registering at least 2.5 on the Richter Scale and having an epicenter within 40 miles of downtown Memphis follows a Poisson distribution with mean 6.5. a)what is the probability that at least 9 such earthquakes will strike next year? Answer 0.208(a) b) Use a suitable approximation to calculate the same probability as in part a. Answer 0.218(b) Approximating the Binomial with a Normal Conditions : np and npq > 5 Approximating the Poisson with a Normal Conditions: > 10 Approximating the Binomial with a Poisson Conditions: n > 20 and p < 0.05