It is claimed that the function is the joint 16
distribution function of the random variables X and Y. Then:
i) determine the corresponding joint probability density function fx,y and
ii) calculate the probability
दिए गए joint distribution function: \[ F_{x,y} = \frac{1}{16}xy(x+y), \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 2 \] ### (i) Joint Probability Density Function (PDF) निकालना PDF प्राप्त करने के लिए Cumulative Distribution Function (CDF) का आंशिक रूप से विभेदन (partial differentiation) करना होगा: \[ f_{x,y} = \frac{\partial^2}{\partial x \partial y} F_{x,y} \] पहले \( x \) के अनुसार _______ ____ ________ __________ ______ ___ __________.
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What is branching process? Give two real examples of branching process.
If )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.
random variables and the random variables
Then show that:and
For the (M/M/K) : (∞ / FIF )0 queuing model with arrival rate λ and service rate per service channel µ, obtain the steady-state probability of n customers in the system, P . n Hence or otherwise, find the probability that an arrival has to wait.
State whether the following statements are true or false. Justify your answer with a short proof or a counter example:
i) Although in one-step transition probability matrix, ,P of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, . This rule is not necessary.
ii) For the joint pdf of random variables (X,Y) given by:
iii) Let the r.v. X follows the negative exponential distribution with parameterThen
iv) One of the examples of non-Poisson queuing system is the (M / G (:)1/ ∞ (FIF0) queuing model.
v) be a random sample from
then maximum likelihood estimators o
The certain item is manufactured by three factories, say 1, 2 and 3. It is known that 1 turns out twice as many items as 2 and that 2 and 3 turn out the same number of items (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective while 4 percent of those manufactured by 3 are defective. All the items produced are put into one stock pile and then one item is chosen at random. The chosen item was found defective. What is the probability that it was produced in factory 1?
The certain item is manufactured by three factories, say 1, 2 and 3. It is known that 1 turns out twice as many items as 2 and that 2 and 3 turn out the same number of items (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective while 4 percent of those manufactured by 3 are defective. All the items produced are put into one stock pile and then one item is chosen at random. The chosen item was found defective. What is the probability that it was produced in factory 1?
Let {Xn; }0 n ≥ be a Markov chain with four states; 1, 2, 3, 4 and the following transition probability matrix:
| 1 | 2 | 3 | 4 | |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 1/2 | 1/2 | 0 | 0 |
| 4 | 1/3 | 1/3 | 1/3 | 0 |
It is claimed that the function is the joint 16
distribution function of the random variables X and Y. Then:
i) determine the corresponding joint probability density function fx,y and
ii) calculate the probability