Warm Up 11.16.14 The most common question asked was ‘why does the variance have to be equal to the mean?’ so let’s prove it! First you will need to explain why the following is true: Use the above to demonstrate that E(X2) = λ2 + λ Now that you have the parts finish your proof that Var(X) = λ

P(X = 1) + P(X = 3) = P(X = 0) + P(X = 2) Warm Up #2 The random variable X follows a Poisson distribution with mean m and satisfies P(X = 1) + P(X = 3) = P(X = 0) + P(X = 2) Find the value of m correct to four decimal places. For this value of m, calculate P(1 ≤ X ≤ 2).

Test Next Class! 5.2 SWBAT… Concepts of trial, outcome, equally likely outcomes, sample space (U) and event; the probability of event A as P(A)=n(A)n(U) ; complementary events A and A’; Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems. 5.3 SWBAT… determine probabilities for combined events, understand and utilize the formula for P(A⋃B), determine mutually exclusive events. 5.4 SWBAT… calculated conditional probabilities, probabilities for independent events and use Bayes’ theorem for a maximum of three events. 5.5 SWBAT… Understand concepts of discrete and continuous random variables and their probability distribution. Define and use probability density functions. Find expected values (mean), mode, median, variance and standard deviation. Apply random variables and expected value. 5.6 SWBAT… Analyze situations by applying the binomial distribution and examining the mean and its variance, apply Poisson distribution and utilize its mean and variance 5.7 SWBAT… Understand the Normal distribution, its properities and the standardized normal values

Homework – IA’s!!! Come prepared with IA topic and why you are interested in the topic If still trying to figure out what you want to do then come with a list of things you are personally interested in

Normal Answers P.2 68-95-99.7 a. 16% b. 84% c. 99.7% d.0.15% X~N(184,5) PDF finds exact value, CDF finds range P.3 a. Physics: -0.463 Chem: 0.431 Maths: 0.198 German: 0.521 Bio: -0.769 b. G, C, M, P, B 2) Inverse does the reverse process, takeing you from the % to the value 3)

Standardized Normal Z-Score: # of standard deviations x is from the mean Z~N(0,1)

Inverse Normal When to use??? How to do I use a calculator…? P(X<k) vs P(X>k)

Inverse Normal Use when given a probability and are asked to calculate corresponding measurement How to do I use a calculator…? P(X<k) vs P(X>k) Calculator gives Probability for area left (meaning < ) of k

Example 1 A university professor determines that no more than 80% of this year’s History candidates should pass the final examination. The examination results were approximately normally distributed with mean 62 and standard deviation of 13. Find the lowest score necessary to pass the exam. Draw the normal curve that illustrates the situation X~N(62, 13^2) P(x>k)=.8 P(x<k)=.2 K=51.059  52 needed to pass

Example 2 Seth is studying O-Chem and Economics. He sits for the mid-year exams in each subject. His O-Chem mark is 56% and the class mean and standard deviation are 50.2% and 15.8% respectively. In Economics he is told that the class mean and standard deviation are 58.7% and 18.7% respectively. What percentage does Seth need to score in Economics, to have an equivalent result to his O-Chem mark? Draw the normal curve that illustrates the situation 65.6%

Poisson Distribution What is it? What is it for? Equation? Variables? Mean and Variance?

Poisson Distribution Answers Most widely used and applied distribution to real world situations. Allows you to count the number of occurrences over a period or range (not necessarily time). Large # of potential emailers each w/ small probability of sending email Area of land able to get earthquakes, small probability of earthquake at any given moment Answers vary. Examples? See Last Class 22.3% 6.564% 22.38% 35 9.88% See Handout for remaining solutions

Distributions Normal Binomial Poisson When to use

Review Events A and B are such that P(A) = 0.3 and P(B) = 0.4. (a) Find the value of P(A B) when (i) A and B are mutually exclusive; (ii) A and B are independent. (b) Given that P(A B) = 0.6, find P(A | B).

Review The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and a standard deviation of 0.2 kg. (a) Determine the probability that a fish that is caught weighs less than 1.4 kg. (b) John catches 6 fish. Calculate the probability that at least 4 of the fish weigh more than 1.4 kg. (c) Determine the probability that a fish that is caught weighs less than 1 kg, given that it weighs less than 1.4 kg.

Review Find the probability of getting a pair on your first roll in Yahtzee

Review The ten numbers x1, x2, ..., x10 have a mean of 10 and a standard deviation of 3. Find the value of

Review A biased coin is weighted such that the probability of obtaining a head is . The coin is tossed 6 times and X denotes the number of heads observed. Find the value of the ratio

Review After being sprayed with a weedkiller, the survival time of weeds in a field is normally distributed with a mean of 15 days. (a) If the probability of survival after 21 days is 0.2, find the standard deviation of the survival time. When another field is sprayed, the survival time of weeds is normally distributed with a mean of 18 days. (b) If the standard deviation of the survival time is unchanged, find the probability of survival after 21 days.

Review After a shop opens at 09:00 the number of customers arriving in any interval of duration t minutes follows a Poisson distribution with mean . (a) (i) Find the probability that exactly five customers arrive before 10:00. (ii) Given that exactly five customers arrive before 10:00, find the probability that exactly two customers arrive before 09:30. (b) Let the second customer arrive at T minutes after 09:00. (i) Show that, for t > 0, P(T > t) = (ii) Hence find in simplified form the probability density function of T. (iii) Evaluate E(T). (You may assume that, for n + and a > 0, .)

Journal Explain the purpose that the r! serves in the Combination formula.