b) Verify Euler’s theorem for the function
Nice — let’s verify Euler’s theorem for $$ f(x,y)=\frac{x+y}{\,x-y\,}. $$ Step 1 — degree. Check homogeneity: for any $t\neq0$, $$ f(tx,ty)=\frac{tx+ty}{tx-ty}=\frac{x+y}{x-y}=f(x,y), $$ so $f$ is homogeneous of degree ________ ______ __________ ____ _____ ______ _______ ___ ______.
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a) A and B are two events which are independent. The probability that both A and B occur is and the probability that neither of them occurs is
Find the probability of the occurrences of A and B.
b) A bag contains 3 white balls and 2 red balls. Another bag contains 5 white and 3 red balls. A bag is chosen at random and a ball is drawn from it. Find the probability that it is white.
c) Two groups of 10 plants each were grown on two different fertilizers. The average height of first group of plants was 92.44 cm, with a standard deviation of 4 cm. For the second group, the average height was 90 cm with a standard deviation of 2 cm. At 5% level of significance test the hypothesis that first fertilizer is better than the second in terms of the plant growth.
c) The difference of mean and variance of a binomial distribution of 9 trails is 4. Find the probability p of success of the binomial distribution. Also find the probability of (i) exactly two successes and (ii) less than 2 successes.
a) It is known that 10 men out of every 100 men and 30 women out of every 1000 women are color blind. In a community, half the population is male. Using Baye’s theorem find the probability that a color blind person chosen at random from among all color blind persons in the community is male.
State whether the following statements are true or false giving reasons in support of your answer.
i) A binomial distribution has mean 3 and variance 4.
ii) The function has no maxima or minima.
iii) The line of regression of x on y is the same as the line of regression of y on x .
iv) The plane is parallel to the line
v) A continuous random variable can have probability density function
c) A club has 9 member having ages 21, 28, 23, 29, 52, 43, 32, 37 and 30 years. One has to be at least 30 years of age to be eligible for the presidentship of the club. A simple random sample of size 5 is selected to provide an estimate of the population proportion eligible for presidentship. Find the mean and the standard error of this estimate