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- 1.9.5. Let a random variable $X$ of the continuous type have a pdf $f(x)$ whose graph is symmetric with respect to $x=c .$ If the mean value of $X$ exists, show that $E(X)=c$Hint: Show that $E(X-c)$ equals zero by writing $E(X-c)$ as the sum of two integrals: one from $-\infty$ to $c$ and the other from $c$ to $\infty .$ In the first, let $y=c-x$ and, in the second, $z=x-c .$ Finally, use the symmetry condition $f(c-y)=f(c+y)$ in the first.Suppose that x has a hypergeometric distribution with N = 300, n = 30, and K=100 a. Find p(x=10) by using the hypergeometric distribution b. Find p(x=10) by using the approximate binomial distribution c. Is the binomial approximation reasonable?The time for the first widget to be manufactured each morning is random between 1 and 3 seconds with a pdf of f(x)=1/162(x-3)^2(x+6) for 0 < x < 6What is the cumulative distribution function, F(x)? What is the probability of a widget being made earlier than 4 s?Answer to four decimal places What is the probability of a widget being made later than 1.3 s?Answer to four decimal places
- Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?they take samples of 4 fireworks for quality co from and examine them for defects. let X be the number of defective fireworks in the sample of 4.A CI is desired for the true average stray-load loss ? (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with ? = 2.2. (Round your answers to two decimal places.) (a) Compute a 95% CI for ? when n = 25 and x = 53.7. , watts(b) Compute a 95% CI for ? when n = 100 and x = 53.7. , watts(c) Compute a 99% CI for ? when n = 100 and x = 53.7. , watts(d) Compute an 82% CI for ? when n = 100 and x = 53.7. , watts(e) How large must n be if the width of the 99% interval for ? is to be 1.0? (Round your answer up to the nearest whole number.)n = You may need to use the appropriate table in the Appendix of Tables to answer this question.
- In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Suppose that x has a hypergeometric distribution with N = 300, n = 30, and K=100. 1. Find p(x=10) by using the hypergeometric distribution. 2. Find p(x=10) by using the approximate binomial distribution. 3. Is the binomial approximation reasonable?
- Consider a random sample X1,...,Xn (n > 2) from Beta(θ,1), where we wish to estimate the parameter θ. (a) Find the MLE θˆ and write it as a function of T = − ∑ni=1 log Xi. (b) Find the sampling distribution of T = − ∑ni=1 log Xi . (Hint: First find the distribution of Ti = − log Xi .)Suppose that X and Y have a discrete joint distribution for which the joint probability mass function is defined as follows: pX,Y(k,l)= c|k+l|, for k=1,0,1,and l=1,0,1; 0, otherwise. (a) Find c. (b) Find the marginal probability mass function pX(·) for X. (c) Compute P(|X - Y | less than or equal 0.9).Suppose that 2 balls are randomly selected from an urn containing 3 red, 4 blue, and 5 white balls. If we let X and Y denote, respectively, the number of red and white balls chosen . Hint: This is an extension of Hypergeometric distribution. Round off your answer into number with 4 decimal places (for example 0.9452). Solve for P(X=1, Y=2).