代做550.420/620/421 Probability - SPRING 2023 MIDTERM 2代做Statistics统计
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MIDTERM 2
1. Suppose the temperature X (in degrees Celsius) is N(10, 25).
(a) Find the values of P (0 < X < 15) and P (X ≥ 20) in terms of the CDF Φ(z) of astandard normal. Clearly label your answers.
(b) If we convert X to degrees Fahrenheit via the formula Y 5/9X + 32, is Y normal? If not why? If so compute the mean and variance of Y.
2. Compute E(X) and Var(X) for the random variable with moment-generating function
Also, determine the probability distribution of X (either its pmf or pdf).
3. If X ~ Beta(Q; β) distribution with Q > 1, compute E(X/1− X) and explain why we require α > 1.
4. These are separate questions.
(a) Someone flips a fair coin three (3) times. Let X be the number of heads in the first two flips, let Y be the number of heads in the last two flips.
Construct the joint probability mass function in tabular form. please.
(b) Give an example of a random variable X that has finite mean and such that E[X2 ] ≤ E[X]. Be explicit by showing that the distribution you selected satisfies the condition.
(c) Give an example of three random variables X, Y, Z in the same family of distributions (e.g., Bernoulli, Exponential, etc.) that when independent have the property that X + Y has the same distribution as aZ + b for some constants a and b.
5. Suppose X ~ exp(a), where a > 0 is a constant. Derive the pdf of Y = eaX by CDF method.
6. We have two random variables X and Y sitting on a table: X ~ unif(0, 1) and Y ~ Bernoulli(2/1).
(a) If you know them, write down the CDF for each rv. Otherwise, derive the one(s) you don’t know. Clearly label your answers. Be sure to state the domains of definition.
(b) We uniformly at random pick one of the two rvs, i.e., we pick X or Y with probability 2/1 each. Let W be the rv that results. Compute the CDF FW (w) = P (W ≤ w) of W , and graph it below.
7. Many discrete random variables take values only in the set of nonnegative integers: 0 , 1, 2, 3, . . . . For such rvs we can define the probability generating function (PGF) for -1 ≤ θ ≤ 1 by
(a) Compute the PGF for exactly one of these (you can choose which, but only one!):
X ~ Poisson(λ) or X ~ binom(n, p) or X ~ geom(p).
(b) (separate question) If X and Y are independent and nonnegative integer-valued, show whether or not gX+Y(θ) = gX (θ)gY (θ). Was independence needed? If so, where? Please be clear.
8. Consider the (concave down) random quadratic polynomial
g(x) = -x2 + Bx + 1;
where B ~ N(1; 1). Compute the expectation of the (global) maximum value of this polynomial.
Hint: First find the maximizer, i.e., the point x* where g(x) attains its maximum value, as a function of the random variable B . Then the (global) maximum is g(x* ).
9. Suppose X ~ Poisson(λ) and Y ~ geom(p) are independent:
for x = 0, 1, 2, 3, . . . ; and P(Y = y) = p(1 − p) y−1 for y = 1, 2, 3, . . . .
Compute P (X = Y - 1).
10. Let X ~ exp(1), and consider the rv Y defined as
Derive the pdf of Y .