1. Suppose that the random variables X and Y have the joint probability distribution f(r, y) given as:_X \Y10.050.050.100.050.100.3530.000.200.10Find the following:(a) P(X = 2|Y =3)(b) E(X)(c) E (Y)(d) V (X)(e) V (Y)(F) E (X – 2Y)2. Suppose the scores X in some SAT Mathematics exam are normally distributed with a mean score of 500 and astandard deviation of 100.(a) Estimate the percentage of student who took that test and scored between 350 and 760.(b) Estimate the percentile rank of a student who took that test and scored 680.(c) Find P (X – 500|< 75).(d) Find the constant é such that P(X – 500|2) = 0.4.

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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b) Suppose that a continuous random variable X is associated with the following function: (i) Show that c = =1/15 2 71 Page 0 < x < 2, Scx, f(x) = {0, elsewhere. lo, (ii) Calculate the mean and variance of X. (iii) Find the CDF of the random variable X, F(x). (iv) Use either the probability distribution function (pdf) of X, f(x), or the cumulative probab distribution function (cdf) of X, F(x), to compute P(0.25 < x < 1.25).
Q. 2 For the exponential random variable X with parameter à = 2, compute the ollowing probabilities. (a) P(1 5). (c) P(X > 5|X > 3). (d) P(X > 2|X > 3).
9
Q3. A random variable X is normally distributed with µ= 50 and o² = 25 . Find the %3D Probability (a) that it will fall between (i) 0 and 40 (ii) 55 and 100 (b) that it will be (i) large than 54 (ii) smaller than 57
Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(18)=0.4F(25)=0.44F(33)=0.5F(41)=0.53F(46)=0.56F(52)=kF(57)=0.64F(65)=0.67 Assuming that Pr[25<X≤52]=0.17, determine the value of k.
A Consider that the variables X and Y have the following joint probability distribution. 1. 3. 0.05 0.05 0.00 0.05 0.10 0.20 0.10 0.35 0.10 (a) Find the marginal distribution of X (b) Find the marginal distribution of Y (c) Determine if X and Y are statistically independent. (d) Find P(Y = 1|X = 3).
4.23. Suppose that X and Y have the following joint probability function: Find the expected value of g(X, Y) = XY2 . Find µx and µy.
(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.
Let Y denote a random variable with the following cumulative distribution function. if y 1. F(y) : In(4) a. Find the PDF, ƒ(y). b. Evaluate P(Y = 0.5). c. Evaluate P(Y 0.4). d. Evaluate P(Y < 1).
4. Find the mean of the discrete random variable X with the following probability distribution. Use this formula. μ -ΣΙΧ . P (X)] P(x) х. Р(х) x2 x² . P(x) X 14 1 /½ 5. Referring to the table above, solve for the variance and standard deviation. Use this formula. o² = E[x2 . P(X)1 – µ? E[x² - P(X)] – µ² | O =
a and b
Given the following table, which corresponds to the values that a random variable X takes and its probability function p(x): xi 2 3 4 5 6 7 p(xi) 0.088 0.34 0.46 0.06 0.04 0.012 Calculate the distribution function (integral function) of x.

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1. Suppose that the random variables X and Y have the joint probability distribution f(r, y) given as: X \Y 1 3 | 0.05 0.05 0.05 0.10 2 0.10 0.35 3 0.00 0.20 0.10 Find the following: (a) P(X = 2|Y = 3) (b) Е(X) (c) E (Y) (d) V (X) (e) V (Y) ) Е (X — 2Y)