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11)Pea plants contain two genes for seed color, each of which maybe Y (for yellow seeds) or G (for green seeds). Plants that containone of each type of gene are called heterozygous. According tothe Mendelian theory of genetics, if two heterozygous plants arecrossed, each of their offspring will have probability 0.75 ofhaving yellow seeds and probability 0.25 of having green seeds.a) Out of 100 offspring of heterozygous plants, what is theprobability that more than 30 have green seeds? And, whatis the probability that between 30 and 35 inclusive havegreen seeds? [Hint: apply the normal distribution to thebinomial distribution and continuity correction]2Consider a fair coin.a) What is the probability of obtaining exactly 3 heads if a faircoin is flipped 6 times?b) What is the probability of obtaining at least 4 head if a faircoin is flipped 17 times.3Consider a sample x1 ,…, x n and let y i x i c for i 1,2,…, n, where c issome specified number. Give a general argument to show that thesample variance of the y i ‘ s is identical to that of the x i ‘ s . [Hint: Howare y and x related? When trying to find a general argument to showthat the sample variance of the yi ‘ s is identical to that of the x i ‘ s ,2use the variance formula s yyi y 2n 1.]4A random variable x has an exponential distribution with parameter ;that is, its density function isf ( x ) e x for x > 0.Express the interquartile range of the distribution in terms of theparameter . [Hint: You will have to first express the lower and upperquartiles of the distribution in terms of the parameter .] 2

11)Pea plants contain two genes for seed color, each of which maybe Y (for yellow seeds) or G (for green seeds). Plants that containone of each type of gene are called heterozygous. According tothe Mendelian theory of genetics, if two heterozygous plants arecrossed, each of their offspring will have probability 0.75 ofhaving yellow seeds and probability 0.25 of having green seeds.a) Out of 100 offspring of heterozygous plants, what is theprobability that more than 30 have green seeds? And, whatis the probability that between 30 and 35 inclusive havegreen seeds? [Hint: apply the normal distribution to thebinomial distribution and continuity correction]2Consider a fair coin.a) What is the probability of obtaining exactly 3 heads if a faircoin is flipped 6 times?b) What is the probability of obtaining at least 4 head if a faircoin is flipped 17 times.3Consider a sample x1 ,…, x n and let y i x i c for i 1,2,…, n, where c issome specified number. Give a general argument to show that thesample variance of the y i ‘ s is identical to that of the x i ‘ s . [Hint: Howare y and x related? When trying to find a general argument to showthat the sample variance of the yi ‘ s is identical to that of the x i ‘ s ,2use the variance formula s yyi y 2n 1.]4A random variable x has an exponential distribution with parameter ;that is, its density function isf ( x ) e x for x > 0.Express the interquartile range of the distribution in terms of theparameter . [Hint: You will have to first express the lower and upperquartiles of the distribution in terms of the parameter .] 2

11)Pea plants contain two genes for seed color, each of which maybe Y (for yellow seeds) or G (for green seeds). Plants that containone of each type of gene are called heterozygous. According tothe Mendelian theory of genetics, if two heterozygous plants arecrossed, each of their offspring will have probability 0.75 ofhaving yellow seeds and probability 0.25 of having green seeds.a) Out of 100 offspring of heterozygous plants, what is theprobability that more than 30 have green seeds? And, whatis the probability that between 30 and 35 inclusive havegreen seeds? [Hint: apply the normal distribution to thebinomial distribution and continuity correction]2Consider a fair coin.a) What is the probability of obtaining exactly 3 heads if a faircoin is flipped 6 times?b) What is the probability of obtaining at least 4 head if a faircoin is flipped 17 times.3Consider a sample x1 ,…, x n and let y i x i c for i 1,2,…, n, where c issome specified number. Give a general argument to show that thesample variance of the y i ‘ s is identical to that of the x i ‘ s . [Hint: Howare y and x related? When trying to find a general argument to showthat the sample variance of the yi ‘ s is identical to that of the x i ‘ s ,2use the variance formula s yyi y 2n 1.]4A random variable x has an exponential distribution with parameter ;that is, its density function isf ( x ) e x for x > 0.Express the interquartile range of the distribution in terms of theparameter . [Hint: You will have to first express the lower and upperquartiles of the distribution in terms of the parameter .] 2

11)Pea plants contain two genes for seed color, each of which maybe Y (for yellow seeds) or G (for green seeds). Plants that containone of each type of gene are called heterozygous. According tothe Mendelian theory of genetics, if two heterozygous plants arecrossed, each of their offspring will have probability 0.75 ofhaving yellow seeds and probability 0.25 of having green seeds.a) Out of 100 offspring of heterozygous plants, what is theprobability that more than 30 have green seeds? And, whatis the probability that between 30 and 35 inclusive havegreen seeds? [Hint: apply the normal distribution to thebinomial distribution and continuity correction]2Consider a fair coin.a) What is the probability of obtaining exactly 3 heads if a faircoin is flipped 6 times?b) What is the probability of obtaining at least 4 head if a faircoin is flipped 17 times.3Consider a sample x1 ,…, x n and let y i x i c for i 1,2,…, n, where c issome specified number. Give a general argument to show that thesample variance of the y i ‘ s is identical to that of the x i ‘ s . [Hint: Howare y and x related? When trying to find a general argument to showthat the sample variance of the yi ‘ s is identical to that of the x i ‘ s ,2use the variance formula s yyi y 2n 1.]4A random variable x has an exponential distribution with parameter ;that is, its density function isf ( x ) e x for x > 0.Express the interquartile range of the distribution in terms of theparameter . [Hint: You will have to first express the lower and upperquartiles of the distribution in terms of the parameter .] 2

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