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DSC4821 Question 9 [5] A random variable X has the chi-square distribution with n degrees of freedom if X has the same distribution as $X_1^2 + X_2^2 + \dots + X_n^2$ where $X_1, X_2, \dots, X_n$ are independent standard normal random variables. Determine the moment generating function of a chi-square distributed random variable with degree of freedom $n = 5$ and deduce its mean and variance. Show all your calculations. Question 10 [10] Let (X,Y) be a continuous bivariate r.v. with joint pdf: $f_{XY}(x,y) = \begin{cases} e^{-(x+y)}, & x > 0, y > 0, \\ 0, & \text{otherwise.} \end{cases}$ 10.1 Find the joint moment generating function of X and Y. 10.2 Find the joint moments $m_{10}, m_{01}$, and $m_{11}$. Question 11 [10] Let $X_1, \dots, X_n$ be n independent r.v.'s and $X_i > 0$. Let: $Y = X_1 \dots X_n = \prod_{i=1}^{n} X_i$. Show that for large n, the pdf of Y is approximately log-normal. Question 12 [15] A radioactive source emits particles according to a Poisson process with rate two particles per minute. 12.1 What is the probability that the first particle appears after three minutes? 12.2 What is the probability that exactly one particle is emitted in the interval from three to five minutes? 12.3 What is the probability that exactly one particle is emitted in the interval from zero to four minutes and that exactly one particle is emitted in the interval from three to five minutes?

Name: dsc4821 question 9 5 a random variable x has the chi square distribution with n degrees of freedom if x has the same distribution as x12 x22 dots xn2 where x1 x2 dots xn are independen 60426
Uploaded: 2024-09-18T10:24:56-08:00
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Description: dsc4821 question 9 5 a random variable x has the chi square distribution with n degrees of freedom if x has the same distribution as x12 x22 dots xn2 where x1 x2 dots xn are independen 60426

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Question 9
[5]
A random variable X has the chi-square distribution with n degrees of freedom if X has the same distribution as $X_1^2 + X_2^2 + \dots + X_n^2$ where $X_1, X_2, \dots, X_n$ are independent standard normal random variables. Determine the moment generating function of a chi-square distributed random variable with degree of freedom $n = 5$ and deduce its mean and variance. Show all your calculations.

Question 10
[10]
Let (X,Y) be a continuous bivariate r.v. with joint pdf:
$f_{XY}(x,y) = \begin{cases} e^{-(x+y)}, & x > 0, y > 0, \\ 0, & \text{otherwise.} \end{cases}$
10.1 Find the joint moment generating function of X and Y.
10.2 Find the joint moments $m_{10}, m_{01}$, and $m_{11}$.

Question 11
[10]
Let $X_1, \dots, X_n$ be n independent r.v.'s and $X_i > 0$. Let:
$Y = X_1 \dots X_n = \prod_{i=1}^{n} X_i$.
Show that for large n, the pdf of Y is approximately log-normal.

Question 12
[15]
A radioactive source emits particles according to a Poisson process with rate two particles per minute.
12.1 What is the probability that the first particle appears after three minutes?
12.2 What is the probability that exactly one particle is emitted in the interval from three to five minutes?
12.3 What is the probability that exactly one particle is emitted in the interval from zero to four minutes and that exactly one particle is emitted in the interval from three to five minutes?

dsc4821 question 9 5 a random variable x has the chi square distribution with n degrees of freedom if x has the same distribution as x12 x22 dots xn2 where x1 x2 dots xn are independen 60426

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