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 आंशिक विभेदन करें: \[ \frac{\partial}{\partial x} F_{x,y} = \frac{\partial}{\partial x} \left(\frac{1}{16} xy (x ______ ________ _________ ____ ___ _______ _________.
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A community has two police cars, which operate independently of one another. The probability that a specific car will be available when needed is 0.99.
i) What is the probability that neither car is available when needed?
ii) What is the probability that a car is available when needed?
What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.
Distinguish between ‘Age Replacement’ and ‘Block Replacement’ policies. Let the lifetimes Y1 ,Y2 ,...., are independently and identically distributed random variables and follows negative exponential distribution with parameter 5. If lifetimes T > 0 and age replacement policy is to be employed, then:
i) find the mean renewal time and
ii) find the long-run average cost per unit time, given the costs C 4 1 = and C 6 2 = units of money.
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, P(j); j = 2,3,4,...... 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 parameter . Then
iv) One of the examples of non-Poisson queuing system is the (M/G/1): (∞(F1F0) queuing model.
v) If X1,X2,..., X, be a random sample from Np, (μ, Σ), then maximum likelihood estimators of µ and ∑are:
What is branching process? Give two real examples of branching process.
If P(s) and Pn(s) respectively is the probability generating function (pgf) of the i.i.d. random variables and the random variables {Xn}, where Xn+1 =
Then show that:
Pn(s) = Pn-1(P(s))
and Pn(s) = Pn-1(P(s)).
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 |
Consider the following system considering of two switches I and II between two points A and B:
A signal is sent from the point A to point B and is received at B if both the switches I and II are closed. It is assumed that the probabilities of I and II being closed are 0.8 and 0.6 respectively and that P [II is closed] I is closed] = P [II is closed].
Find:
i) The probability that signal is received at B.
ii) The conditional probability that switch I was open, given that the signal was not received at B,
iii) The conditional probability that switch II was open, given that the signal was not received at B.