Question #32 a) What is the probability density function of the sum of two independent random...
Question #32
a) What is the probability density function of the sum of two
independent
random variable, each of which is binomial with parameters n and
p?
b) What is the probability density function of the average of
two independent
random variable, each of which is normal with mean μ and
variance
(σ ^2)?
answer for a) is BIN(2n,p)
answer for b) is N(2μ,2σ^2)
please show the steps. thanks.
Homework Answers
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Question #32
a) What is the probability density function of the sum of two
independent
random...
10. What is the probability density of the sum of two independent random variables, each of...
10. What is the probability density of the sum of two independent random variables, each of which is uniformly distributed over the interval 0, 1]?
The next two problems allow you to express the sum of two independent random as a...
The next two problems allow you to express the sum of two independent random as a precise function of each of their probability mass functions or probability density functions in the case they are each discrete or continuous random variables respectively. These problems are conceptually important because they tell you how to compute the distribution of a random walk (which we will define later) from the distribution of its steps (again, defined later) in a general case. 5. Let X,...
QUESTION 4 Suppose Xis a random variable with probability density function f(x) and Y is a...
QUESTION 4 Suppose Xis a random variable with probability density function f(x) and Y is a random variable with density function f,(x). Then X and Y are called independent random variables if their joint density function is the product of their individual density functions: x, y We modelled waiting times by using exponential density functions if t <0 where μ is the average waiting time. In the next example we consider a situation with two independent waiting times. The joint...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
Let Y1, Y2, , Yn be independent, normal random variables, each with mean μ and variance...
Let Y1, Y2, , Yn be independent, normal random variables, each
with mean μ and variance σ^2.
(a) Find the density function of
f Y(u) =
(b) If σ^2 = 25 and n = 9, what is the
probability that the sample mean, Y, takes on a value that is
within one unit of the population mean, μ?
That is, find P(|Y − μ| ≤ 1). (Round your answer to four decimal
places.)
P(|Y − μ| ≤ 1) =
(c)...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
PART V: Recall that for scalar > 0, the probability density function of an "exponential" random...
PART V: Recall that for scalar > 0, the probability density function of an "exponential" random variable with parameter , is P2; 1) = exp(-x). We have n independent samples 11,..., Ir. Each 21, ..., Iris a scalar. Each ris an "exponential" random variable with parameter A. for which 12) (1 point] What is the maximum likelihood estimator? In other words, what is the value of the derivative of (D;) with respect to X is zero? Show all the steps...
2) Two statistically-independent random variables, (X,Y), each have marginal probability density,...
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
Random variable (20) Z X+Y is a random variable equal to the sum of two continuous...
Random variable
(20) Z X+Y is a random variable equal to the sum of two continuous random variables X and Y. X has a uniform density from (-1, 1), and Y has a uniform density from (0, 2). X and Y may or may not be independent. Answer these two separate questions a). Given that the correlation coefficient between X and Y is 0, find the probability density function f7(z) and the variance o7. b). Given that the correlation coefficient...
. Suppose that Y is a normal random variable with mean µ = 3 and variance...
. Suppose that Y is a normal random variable with mean
µ = 3 and variance σ
2 = 1; i.e.,
Y
dist = N(3, 1). Also suppose that X is a binomial random variable
with n = 2 and p = 1/4; i.e.,
X
dist = Bin(2, 1/4). Suppose X and Y are independent random
variables. Find the expected
value of Y
X. Hint: Consider conditioning on the events {X = j} for j = 0, 1,
2.
8....
10. What is the probability density of the sum of two independent random variables, each of which is uniformly distributed over the interval 0, 1]?
The next two problems allow you to express the sum of two independent random as a precise function of each of their probability mass functions or probability density functions in the case they are each discrete or continuous random variables respectively. These problems are conceptually important because they tell you how to compute the distribution of a random walk (which we will define later) from the distribution of its steps (again, defined later) in a general case. 5. Let X,...
QUESTION 4 Suppose Xis a random variable with probability density function f(x) and Y is a random variable with density function f,(x). Then X and Y are called independent random variables if their joint density function is the product of their individual density functions: x, y We modelled waiting times by using exponential density functions if t <0 where μ is the average waiting time. In the next example we consider a situation with two independent waiting times. The joint...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
Let Y1, Y2, , Yn be independent, normal random variables, each
with mean μ and variance σ^2.
(a) Find the density function of
f Y(u) =
(b) If σ^2 = 25 and n = 9, what is the
probability that the sample mean, Y, takes on a value that is
within one unit of the population mean, μ?
That is, find P(|Y − μ| ≤ 1). (Round your answer to four decimal
places.)
P(|Y − μ| ≤ 1) =
(c)...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
PART V: Recall that for scalar > 0, the probability density function of an "exponential" random variable with parameter , is P2; 1) = exp(-x). We have n independent samples 11,..., Ir. Each 21, ..., Iris a scalar. Each ris an "exponential" random variable with parameter A. for which 12) (1 point] What is the maximum likelihood estimator? In other words, what is the value of the derivative of (D;) with respect to X is zero? Show all the steps...
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
Random variable
(20) Z X+Y is a random variable equal to the sum of two continuous random variables X and Y. X has a uniform density from (-1, 1), and Y has a uniform density from (0, 2). X and Y may or may not be independent. Answer these two separate questions a). Given that the correlation coefficient between X and Y is 0, find the probability density function f7(z) and the variance o7. b). Given that the correlation coefficient...
. Suppose that Y is a normal random variable with mean
µ = 3 and variance σ
2 = 1; i.e.,
Y
dist = N(3, 1). Also suppose that X is a binomial random variable
with n = 2 and p = 1/4; i.e.,
X
dist = Bin(2, 1/4). Suppose X and Y are independent random
variables. Find the expected
value of Y
X. Hint: Consider conditioning on the events {X = j} for j = 0, 1,
2.
8....
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