In Exercises
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Finite Mathematics for the Managerial, Life, and Social Sciences
- Population Genetics In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypesthat is, instances in which the two alleles carried at a particular site on an individuals chromosomes are both the same. For population in which blood-related individual mate, them is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. in general. the frequency of homozygous children from mating of blood-related parents is greater than that for children from unrelated parents Measured over a large number of generations, the proportion of heterozygous genotypesthat is, nonhomozygous genotypeschanges by a constant factor 1 from generation to generation. The factor 1 is a number between 0 and 1. If 1=0.75, for example then the proportion of heterozygous individuals in the population decreases by 25 in each generation In this case, after 10 generations, the proportion of heterozygous individuals in the population decreases by 94.37, since 0.7510=0.0563, or 5.63. In other words, 94.37 of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For mating between siblings 1 can be determined as the largest value of for which 2=12+14. This equation comes from carefully accounting for the genotypes for the present generation the 2 term in terms of those previous two generations represented by for the parents generation and by the constant term of the grandparents generation. a Find both solutions to the quadratic equation above and identify which is 1 use a horizontal span of 1 to 1 in this exercise and the following exercise. b After 5 generations, what proportion of the population will be homozygous? c After 20 generations, what proportion of the population will be homozygous?arrow_forwardConsumer Preference In a population of 100,000 consumers, there are 20,000 users of Brand A, 30,000 users of Brand B, and 50,000 who use neither brand. During any month, a Brand A user has a 20 probability of switching to Brand B and a 5 of not using either brand. A Brand B user has a 15 probability of switching to Brand A and a 10 probability of not using either brand. A nonuser has a 10 probability of purchasing Brand A and a 15 probability of purchasing Brand B. How many people will be in each group a in 1 month, b in 2 months, and c in 18 months?arrow_forwardExercise 3. Let X be a random variable with mean µ and variance o². For a € R, consider the expectation E((X − a)²). a) Write E((X - a)²) in terms of a, μ and σ². b) For which value a is E((X − a)²) minimal? c) For the value a from part (b), what is E((X − a)²)?arrow_forward
- A- Let x be a discrete random variable with probability distribution function f(x)=k( x2 +20) and x= −1,1,2,3. Find the value of k. Find the Variance of X. B- Let x denote a discrete random variable which can take the values −2,0, and 5. Given that the expectation of X is 8/100 and P(X=−2)=8/20 , find P(X=5).arrow_forwardWhat is E(Y)?arrow_forwardExercise 4 Suppose X and Y are independent random variables such that X has uniform (0, 1) distribution, Y has exponential distribution with mean 1. b) E(XY) c) E[(X - Y)²] d) E(X²e²Y)arrow_forward
- A grade in a probability course depends on exam scores X1 and X2. The professor, afan of probability, releases exam scores in a normalized fashion such that X1 and X2 are independentGaussian (μ = 0, σ = √2) random variables. The semester average is X = 0.5(X1 + X2). 1. A student was to earn an A grade if X > 1. What is P(A)?2. To improve the RS (Rating Service) score, the professor decides he should award more A’s. Now you get an A if max(X1, X2) > 1. What is P(A) now?3. The professor found out he is unpopular and decides to award an A if either X > 1 or max(X1, X2) > 1. Now what is P(A)?4. Under criticism of grade inflation from the department chair, the professor adopts a new policy. An A is awarded if max(X1, X2) > 1 and min(X1, X2) > 0. Now what is P(A)? Detailed explanation and calculation would be much appreciated.arrow_forwardFind the Mean of f(x), where f(x) = 6X+6. The probability distribution of a random variable X ; if X = 8, 10, 11, 12, 13 and P(X) = 0.2, 0.25, 0.3, 0.15, 0.1 respectively"arrow_forwardFind Covariancearrow_forward
- Exercise 3. Let X be a random variable with mean u and variance o2. For a E R, consider the expectation E((X – a)²). a) Write E((X – a)²) in terms of a, µ and o2. b) For which value a is E((X – a)²) minimal? c) For the value a from part (b), what is E((X – a)²)?arrow_forwardA5. Give the definitions for strictly- and weakly-stationary random processes.arrow_forwardA probability distribution for a random variable Y is given byP(Y=y)=cy, for y=1,2,4,5and c is a constant. Find E(Y), the expected value of Y. Give your answer as a fraction in its simplest form.arrow_forward
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