Let the probability mass function of X be given as 5. | | P(X-x) a) Find the pmf of Y2 b) Find the pmf ofY X2 0.3 0.60 Let the probability mass function of X be given as 5. | | P(X-x) a) Fin...
Question 4. [5 marksi Let Xbe a random variable with probability mass function (pmf) A-p for -1, 2,... and zero elsewhere (whereq-1-p, 0 <p< (a) Find the moment generating function (mg ofX. C11 (b) Using the result in (a) or otherwise find the expected value and variance of X. C23 (c) Let X, X,., X, be independent random variables all with the pmf fix) above, and let Find the mgf and the cumulant generating function of Y.
Let f(x,y) = exp(-x) be a probability density function over the plane. Find the probabilities: Parta)P( X2 + y2 <a), a > 0, Part b)P(x2 + y2 <a), a > 0.
Let X be a discrete random variable with a probability mass function (pmf) of the following quadratic form: p(x) = Cx(5 – x), for x = 1,2,3,4 and C > 0. (a) Find the value of the constant C. (b) Find P(X ≤ 2).
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
Let X be a discrete random variable with probability mass function p(k) = 1/5, k = 1, 2, . . . , 5, zero elsewhere. (a) Find the moment generating function of X. (b) Use the moment generating function in (a) to determine the convolution of two identical probability mass functions given above. This is identical to asking the probability mass function of X + Y and where X and Y are independent and each has probability mass function given...
2. Let X1 and X2 have the joint pmf p(11, 12) = 232, 21 = 1,2,3 and x2 = 1,2,3, zero elsewhere. Find the joint pmf of Y1 = X1 X2 and Y2 = X2, and then find the marginal pmf of Y.
1. (Hint: This pmf should look familiar) Random variables X and Y have joint probability mass function (IPMI): otherwise. (a) Find Fx,y(x, y), the joint cumulative distribution function (CDF) of X and Y. A graphical repre- sentation is sufficient: probably the best way to do this is to draw the x - y plane and label different regions with the value of Fx,y(x, y) in that region. (b) Let Z = X2 + Y2. Find the probability mass function (PMF)...
Question 1. A Discrete Distribution - PME Verify that p(x) is a probability mass function (pmf) and calculate the following for a random variable X with this pmf 1.25 1.5 | 1.7522.45 p(x) 0.25 0.35 0.1 0.150.15 (a) P(X S 2) (b) P(X 1.65) (c) P(X = 1.5) (d) P(X<1.3 or X 221) e) The mean (f) The variance. (g) Sketch the cumulative distribution function (edf). Note that it exhibits jumps and is a right continuous function.
1. (Lec 6 & 7 discrete R.V., 16 pts) The pmf (probability mass function) of a random variable X is shown below: -2 0.2 Let A be the event that X is less than 0. .ן 0.4 otherwise px(x) 0.1 0.1 (a) Find the value of the constant a nd ElI and omai pmf of X given A (c) Find the conditional pmf of X given A. (d) Find E(X[A] and Var[X[A]. (e) Let Y2X 3. Find the pmf of...
Find the density function of Y2x+8 9. Let R have probability mass function (pmf) pr)-1/8 for r1,8 Find (I)the cumulative distribution function (cdf) of R; (2)P(R>5): (S)EI(R-3)(R-)) (6)Var(R 10 Suppose the density function of a random variable X is f(x)sige 2- x > 0, where σ>0 is constant. Find E(X) and D()