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Ex-H) I found out that skewness is defined as 1. For this pmf 3 57 x)...
Question 3 [20 marks] A measure of skewness is defined by Y such that: Note that...
Question 3 [20 marks] A measure of skewness is defined by Y such that: Note that when a distribution is symmetrical about the mean, the skewness is equal to zero. If it is skewed to the right, the measure of skewness will be positive; if it is skewed to the left, the measure of skewness will be negative. Let X be random variable, and a function f(x) is defined such that 21, 2, 3, 4, 5, 6, f(x)- (a) Create...
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete)...
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete) and Y (continuous), for x = 1, 2, 3, 4 and 0 <y < 2. (a) Determine the marginal pmf of X. (b) Determine the marginal pdf of Y. (c) Compute P(X<2 and Y < 1). (d) Explain why X and Y are dependent without computing Cou(X,Y).
13. Let X and Y be rv's whose joint PMF is given by: Y=1 2 3...
13. Let X and Y be rv's whose joint PMF is given by: Y=1 2 3 X=0 0.2 0.1 0 1 0.1 0.3 0. 2 . 0 0 0.3 Compute the covariance and correlation matrix of the random vector (X,Y).
2. Let X and X be two random variables with the following joint PMF Yix 2...
2. Let X and X be two random variables with the following joint PMF Yix 2 0 2 0 0.1 0.05 0.05 0.15 0.1 0.05 0.1 0.05 0.05 0.05 4 0.05 0.05 0.02 0.1 0.03 total 0.2 0.2 0.12 0.3 0.18 total 0.45 0.3 0.25 1 1) Find E[X] and E[Y]. (10 points) 2) What is the covariance of X and Y? (20 points) 3) Are X and Y independent? Explain. (10 points)
Let X be a discrete random variable with the following pmf 0.1 for I = 0.2...
Let X be a discrete random variable with the following pmf 0.1 for I = 0.2 0.2 for x = 0.4 0.2 for x = 0.5 P(X = x) = 0.3 for x = 0.8 0.2 for x = 1 0 otherwise Note: Write your final answers as decimals Find the following a) P(0.25 < X < 0.75) = b) P(X = 0.2|X<0.6) c) E(2X+1) =
Consider a random variable X with RX = {−1, 0, 1} and PMF P(X = −1)...
Consider a random variable X with RX = {−1, 0, 1} and PMF P(X =
−1) = 1/4 , P(X = 0) = 1/2 , P(X = 1) = 1/4 .
a) Determine the moment-generating function (MGF) MX(t) of
X.
b) Obtain the first two derivatives of the MGF to compute E[X]
and Var(X).
Consider a random variable X with Rx = {-1,0,1} and PMF Determine the moment-generating function (MGF) Mx(t) of X b) Obtain the first two derivatives of...
3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2...
3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2 for 0.2 for x=3 Pg(x)=〈 0.1 for x=4 0.25 for x=5 0.15 for x=6 otherwise a) (10 points) Find E[X] b) (10 points) Find Var(X) c) Let Y-* I. (15 points) Find E[Y] II. (15 points) Find Var(Y) X-HX 4. Consider a discrete random variable X with E [X]-4x and Var(X) = σ. Let Y a. (10 points) Find E[Y] b. (20 points) Find...
Consider a random variable X with PMF Px(X) ={ cx x = 1,2,...,5 I found the...
Consider a random variable X with PMF Px(X) ={ cx x = 1,2,...,5 I found the value of c to be 1/15 1. find the complete expression for the CDF of X. 2. Find E[X], the expected value of X.
(5 points) Suppose the joint probability mass function (pmf) of integer- Y ī PlX = í,ys j) = (i + 2j)o, for 0 í valued random variables X and < 2,0 < j < 2, and i +j < 3, where c is a con...
(5 points) Suppose the joint probability mass function (pmf) of integer- Y ī PlX = í,ys j) = (i + 2j)o, for 0 í valued random variables X and < 2,0 < j < 2, and i +j < 3, where c is a constant. In other words, the joint pmf of X and Y can be represented by the table: Y=2 |Y=0 Y=1 X=0| 0 2c 4c 3c 4c 5c X=21 2c (a) Find the constant c. (b) Compute...
Problem 8.2 X Y Discrete random variables X, Y have joint pmf given in the table...
Problem 8.2 X Y Discrete random variables X, Y have joint pmf given in the table to the right, where X takes values in {1,2,3,4} and Y takes values in {1,2,3). 2 3 1 2 3 0. 100.3 0 0.2 0.1 0 0.05 0.1 0 0.1 0.05 (e) Compute the MAP estimate of X given the observation Y = 2. Compute the posterior probabiity of error of this estimate, given that Y = 2. (f) Compute the MMSE estimate of...
Question 3 [20 marks] A measure of skewness is defined by Y such that: Note that when a distribution is symmetrical about the mean, the skewness is equal to zero. If it is skewed to the right, the measure of skewness will be positive; if it is skewed to the left, the measure of skewness will be negative. Let X be random variable, and a function f(x) is defined such that 21, 2, 3, 4, 5, 6, f(x)- (a) Create...
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete) and Y (continuous), for x = 1, 2, 3, 4 and 0 <y < 2. (a) Determine the marginal pmf of X. (b) Determine the marginal pdf of Y. (c) Compute P(X<2 and Y < 1). (d) Explain why X and Y are dependent without computing Cou(X,Y).
13. Let X and Y be rv's whose joint PMF is given by: Y=1 2 3 X=0 0.2 0.1 0 1 0.1 0.3 0. 2 . 0 0 0.3 Compute the covariance and correlation matrix of the random vector (X,Y).
2. Let X and X be two random variables with the following joint PMF Yix 2 0 2 0 0.1 0.05 0.05 0.15 0.1 0.05 0.1 0.05 0.05 0.05 4 0.05 0.05 0.02 0.1 0.03 total 0.2 0.2 0.12 0.3 0.18 total 0.45 0.3 0.25 1 1) Find E[X] and E[Y]. (10 points) 2) What is the covariance of X and Y? (20 points) 3) Are X and Y independent? Explain. (10 points)
Let X be a discrete random variable with the following pmf 0.1 for I = 0.2 0.2 for x = 0.4 0.2 for x = 0.5 P(X = x) = 0.3 for x = 0.8 0.2 for x = 1 0 otherwise Note: Write your final answers as decimals Find the following a) P(0.25 < X < 0.75) = b) P(X = 0.2|X<0.6) c) E(2X+1) =
Consider a random variable X with RX = {−1, 0, 1} and PMF P(X =
−1) = 1/4 , P(X = 0) = 1/2 , P(X = 1) = 1/4 .
a) Determine the moment-generating function (MGF) MX(t) of
X.
b) Obtain the first two derivatives of the MGF to compute E[X]
and Var(X).
Consider a random variable X with Rx = {-1,0,1} and PMF Determine the moment-generating function (MGF) Mx(t) of X b) Obtain the first two derivatives of...
3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2 for 0.2 for x=3 Pg(x)=〈 0.1 for x=4 0.25 for x=5 0.15 for x=6 otherwise a) (10 points) Find E[X] b) (10 points) Find Var(X) c) Let Y-* I. (15 points) Find E[Y] II. (15 points) Find Var(Y) X-HX 4. Consider a discrete random variable X with E [X]-4x and Var(X) = σ. Let Y a. (10 points) Find E[Y] b. (20 points) Find...
(5 points) Suppose the joint probability mass function (pmf) of integer- Y ī PlX = í,ys j) = (i + 2j)o, for 0 í valued random variables X and < 2,0 < j < 2, and i +j < 3, where c is a constant. In other words, the joint pmf of X and Y can be represented by the table: Y=2 |Y=0 Y=1 X=0| 0 2c 4c 3c 4c 5c X=21 2c (a) Find the constant c. (b) Compute...
Problem 8.2 X Y Discrete random variables X, Y have joint pmf given in the table to the right, where X takes values in {1,2,3,4} and Y takes values in {1,2,3). 2 3 1 2 3 0. 100.3 0 0.2 0.1 0 0.05 0.1 0 0.1 0.05 (e) Compute the MAP estimate of X given the observation Y = 2. Compute the posterior probabiity of error of this estimate, given that Y = 2. (f) Compute the MMSE estimate of...
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