1. Suppose X and Y are a bivariate continuous random variable with 0sXs2 and 0SYS3 and have a valid joint PDF: f(x.y) = c (x2 + y12). Rounded to TWO decimal places the value of c is and the conditional probability Prí1
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- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)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?
- Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
- If Y is a continuous, uniformly distributed random variable over the interval(4,10), then the value of the PDF between 4 and 10 is?The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1Q4) If X is a continuous random variable having pdf ke~ (2x+3y) x>0y>0 xy) = = e p(x) { 0 otherwise Find a) the constant k b) P(X>1) ¢) X, X2, 02, standard deviation.
- 10 Suppose that X is a continuous random variable with pdf given by X(x) =cxα−1e−(x/β)α for x≥0, where α >0,β >0 are some parameters of the distribution, and c is a constant. (a) Find the value of c such that fX is a valid pdf.(b) Find the cdf and the quantile function of X.(c) Ifα= 2 andβ= 4, calculate P(X <1) and find the quartiles of X.Suppose that two random variables X and Y have the joint PDFfXY(u, v) = {60(u^2)v u ≥ 0, v ≥ 0, and u + v ≤ 1, 0 otherwise.(a) Are X and Y independent?(b) What is the marginal distribution fX(t)?(c) What is Pr[X ≥ Y]? (To set up the right integral, it might help you to draw the rangeof (X, Y) in the uv-plane and identify the region within that range where u ≥ v.)In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signicantly dierent from$21, which is the average bill for current AT&T customers (d) Using 5% signicance level, calculate the critical values of the test. Is the null hypothesis rejected?(e) The competitor would react by using dierent set of hypotheses. For example, consider H0 : µ = 19 andH1 : µ 6= 19. What…