Basics of Probability

Transcription

1 Basics of Probability By A.V. Vedpuriswar October 2, , 2016

2 Random variables and events A random variable is an uncertain quantity. A outcome is an observed value of a random variable. An event is a single outcome or a set of outcomes. Mutually exclusive events cannot happen at the same time. Exhaustive events include all possible outcomes. Independent events refer to events for which the occurrence of one has no influence on the occurrence of others. Events which are not independent, are called dependent.

3 Discrete and continuous random variables A discrete random variable is one for which the number of possible outcomes can be counted. For each possible outcome, there is a measurable and positive probability. A continuous random variable is one for which the number of possible outcomes is infinite. For a continuous random variable, probability is measurable for a range of data points but not for a specific point.

4 Binomial and Uniform random variables A binomial random variable can be defined as the number of successes in a given number of trials where the outcome can be a success or failure. The probability of success is constant for each trial and the trials are independent. If there are n trials, with probability of success in each trial is p, he probability of getting r successes = ncr (p)^r (1-p)^(n-r) r A discrete uniform random variable is one for which the probabilities for all possible outcomes for a discrete random variable are equal. The continuous uniform distribution is defined over a range (a, b) such that p (x 1 x 2 ) = (x 2 - x 1 ) / (b-a)

5 Types of probability Type of probability Empirical A priori Subjective Unconditional Conditional Joint Description Based on data Based on reasoning/inspection Based on personal assessment Probability of an event regardless of the occurrence of other events Probability of an event influenced by the occurrence of other events Probability that both events will occur. 4

6 Univariate and multivariate distributions A univariate distribution is the distribution of a single random variable. A multivariate distribution specifies the probabilities associated with a group of random variables.

7 What is the probability of drawing an Ace or a Spade from a deck of cards? Solution No. of Aces = 4 No. of Spades = 13 Total no of Aces and Spades = 17 Less: Ace of Spades = 1 = 16 Probability = 16C 1 / 52C 1 =16 52 = 4 13

8 In a selection process, 30 candidates will qualify finally. 600 appear in the written test and 100 will be called for interview. What is the probability that a person writing the test will be called for interview? Determine the possibility of a person being selected if he has been called for interview. Solution Probability of being called for interview = =1 6 Probability of being selected if called for interview = = 3 10 Probability of a person appearing in the test being finally selected = 1 6 x 3 10 = 1 20

9 80% of all tourists who come to India visit Delhi, 70% of them visit Mumbai and 60% visit both. What is the probability that the tourists will visit Mumbai or Delhi or both? Solution Mumbai Delhi Probability of visiting Delhi or Mumbai or both = = 90%

10 When two six sided dice are tossed, what is the expected value of the sum of the faces? Outcome Sum No. of ways (1,1) 2 1 (1,2), (2,1) 3 2 (1,3), (2,2), (3,1) 4 3 (1,4), (2,3), (3,2), (4,1) 5 4 (1,5), (2,4), (3,3), (4,2) (5,1) 6 5 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 7 6 (2,6), (3,5), (4,4), (5,3), (6,2) 8 5 (3,6), (4,5), (5,4), (6,3) 9 4 (4,6), (5,5), (6,4) 10 3 (5,6), (6,5) 11 2 (6,6) Expected value of sum =(1)(2)+(2)(3)+(3)(4)+(4)(5)+(5)(6)+(6)(7)+(5)(8) + (4)(9) + (3)10) + (2)(11) + (1)(12) 36 = ( ) / 36 = 7

11 A fair coin is tossed 5 times. What is the probability that it lands up tail at least once? Solution Probability of landing a head each time = 1 2 Probability of getting a head all 5 times = (1 2) 5 =1 32 Probability of getting at least one tail = = 31 32

12 There are ten sprinters in the Olympic finals. How many different ways can the gold, silver, bronze medals be awarded? Assume dead heat is not possible. Solution Gold can be won in 10 ways Silver can be won in 9 ways Bronze can be won in 8 ways So total no. of ways = 10 x 9 x 8 = 720 ways

13 From a group of 6 men and 4 women, a committee of 4 is to be chosen. What is the probability that the committee consists exactly of two men and two women? Solution Four people can be chosen in 10C 4 ways If we want exactly two men and two women, we can choose in 6C 2 x 4C 2 ways probability = [6C 2 x 4C 2 ] /10C 4 = (15 X 6) 210 = 3 7

14 25% of all households in a town have broadband Internet access. In a random sample of 15 houses, what is the probability that exactly 5 have Internet access? Solution Probability of having internet access =.25 Probability of not having access =.75 Required probability = 15C 5 X (0.25) 5 X (0.75) 10

15 There are 10 bonds in a portfolio. The probability of default for each of the bonds over the coming year is 5%. These probabilities are independent of each other. What is the probability that exactly one bond defaults? Required probability = 10C 1 (.05)(.95) 9 =.3151 = 31.51% 14

16 A CDS portfolio consists of 5 bonds with zero default correlation. One year default probabilities are : 1%, 2%, 5%,10% and 15% respectively. What is the probability that that the protection seller will not have to pay compensation? Probability of no default = (.99)(.98)(.95)(.90)(.85) =.7051 = 70.51% 15

17 The 5 year cumulative probability of default for a bond is 15%. The marginal probability of default for the sixth year is 10%. What is the six year cumulative probability of default? Probability of no default = (.85)(.9) Required probability = 1- (.85)(.90) = = 23.5% 16

18 There is a 60% probability that the economy will outperform. If it does, there is a 70% chance a stock will be up and 30% chance it will go down. There is a 40% probability that the economy will underperform and if it does, there is a 20% chance that the stock will increase in value and 80% chance that it will not. Given that the stock has gained, what is the probability that the economy has outperformed? Probability that the stock will gain and the economy will outperform = (.6) (.7) =..42 Probability that the stock will gain and the economy will underperform = (.4) (.2) =.08 Probability that the economy has outperformed given the stock has gained =.42 /( ) =.42/.50 = 0.84 = 84%

19 Expected value, covariance, correlation Expected value is the weighted average of the possible outcomes of a random variable, the weights being the probabilities associated with different outcomes. Covariance is a measure of how two assets move together. Correlation indicates the strength of a linear relationship between a pair of random variables. Correlation coefficient is the covariance divided by the product of standard deviations of the two variables. Correlation coefficient, = Cov (x,y) / σ x σ y Coefficient of variation = Standard deviation / Mean

20 What is the expected value of a stock given the following information? Price Probability Solution Expected value = (60) (.2) + (65) (.3) + 70) (.3) + 75) (.2) = = 67.5

21 x p (x) What is the variance? Solution Expected value = 2.5 x p (x) (x-x) 2 p (x) (-1.5) 2 (0.2) = (-0.5) 2 (0.3) = ( 0.5) 2 (0.3) = (1.5) 2 (0.2) =.45 = 1.05

22 The correlation of returns between stocks A & B is The covariance between the two securities is and the standard deviation of B is What is the variance of A? Solution Correlation = Cov(A,B) / σ A σ B or.50 = (.0043) / [(σ A ) (.26)] or σ A =.0043 /[0.26 x.5] =.0043 /.130 =.0331 or σ 2 A =.0011

23 You are building a portfolio of three stocks as follows: Stock Weight in portfolio Expected return Standard deviation A B C The correlation coefficients are : P A, B =0.85; P A, C = 0.30; P B, C = What is the expected return and standard deviation of the portfolio?

24 Solution Expected return = (.55) (.08) + (.25) (.04) + (.20) (.03) = =.06 Variance =.55 2 x x x (2) (.55) (.25) (.85) (.24) (.18) + (2) (.25) (.20) (-.15) (.18) (.15) + (2) (.55) (.20) (.30) (.24) (.15) =

25 Given a risk free return of 4%, which portfolio gives the best risk adjusted return? Portfolio A B C D Return 5% 11% 14% 18% Standard deviation 8% 21% 34% 40% We subtract risk free return from the actual return and divide by the standard deviation. Portfolio A : (5-4) / 8 =.125 Portfolio B : (11-4) / 21 =.333 Portfolio C : (14-4) / 34 =.294 Portfolio D : (18-4) / 40 =.35 So D is the best portfolio.