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