Month: June 2022
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a football post?! — Xi’an’s Og
I am not interested in football, neither as a player (a primary school trauma when I was the last being picked!) or as a fan, contrary to my dad (who was a football referee in his youth) and my kids, but Gareth Roberts (University of Warwick) and Jeff Rosenthal wrote a paper on football draws […]…
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JAM 2022 [ 41 -50]
Let [katex] \{a_n \}_{n \geq 1} [/katex] be a sequence of real numbers such that a1+5m=2 , a2+5m=3 , a3+5m=4 , a4+5m=5, a5+5m=6 , m=0,1,2, … . Then equals ____________________________ . It is evident that an is defined for all values of n=1,2,3,…. . And so
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RANDOM NUMBER GENERATORS
What are random numbers? Imagine yourself picking up a card from a well shuffled full deck of cards. What could the card you picked be? The jack of spades, the king of diamonds, or the queen of hearts?? Well it could be the ace of clubs too or it could be any of the other…
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SIMULATION
Simulation is a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical models that describe the behavior of business or economic system or some component thereof over extended period of time. Simulation deals with both abstract and physical models. Some simulation with physical and abstract models might…
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Monte Carlo Methods
A Monte Carlo method is a computational / numerical method that uses random numbers to compute / estimate a quantity of interest. The quantity of interests may be the mean of a random variable , functions of several means , distributions of random variables or high dimensional integrals. Basically, Monte Carlo methods may be grouped…
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Operations Research: The ORIGIN
The origin of Operations Research can be traced back to the World War-IIperiod. During the time of war, the military management in England had calledupon a team of scientists to study the strategic and tactical problems related toair and land defence of the country. Since, the military resources were limited,it was upto these scientists to…
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JAM 2022 [ 51-60 ]
[Q No. 51-60] Let [katex] X_1,X_2,\dots,X_9 [/katex] be a random sample from a [katex] N(\mu,\sigma^2) [/katex] distribution, where [katex] \mu \in \mathbb{R} \text{ and } \sigma>0 [/katex] are unknown. Let the observed values of [katex] \bar{X} = \frac{1}{9} \sum_{i=1}^9 \text{and} S^2 = \frac{1}{8} \sum_{i=1}^{9} ( X_i -\bar{X} )^2 [/katex] be 9.8 and 1.44, respectively. If…
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HYPERGEOMETRIC DISTRIBUTION
Suppose an urn contains N coloured balls, out of which M are blue and the rest are red in color. Suppose we draw a sample of ‘n’ number of balls from the urn at random and without replacements. Then if X a random variable denoting the number of blue balls drawn, then the probability that…
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Survival Analysis: Competing Risk Theory
INTRODUCTION A distinctive feature of survival data is the concept of censoring. And an implicit concept in the definition of censoring is that if the study had been prolonged (or if subjects had not dropped out), eventually the outcome of interest would have been observed to occur for all the subjects. Conventional statistical methods for the…
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STB – 2019
Consider a count variable X following a Poisson distribution with parameter θ > 0, where zero count (i.e., X = 0) is not observable. We have n observations X1, . . . , Xn from this distribution. Let denote the sample mean. a) Derive the quantity for which is an unbiased estimator. b) Suppose that the observed value…