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CSIR NET Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free to read right now. Pick your subject below.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

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Tagged “probability”

“Convergence in probability ⇒ almost sure convergence”
“Uncorrelated ⇒ independent”
“Pairwise independent events are mutually independent”

False — Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree

“Every random variable has a moment generating function”

False — The standard Cauchy distribution

“A mixture of two distributions is a linear combination of the variables”
“The maximum of independent exponentials is exponential”

False — max⁡(X1,X2)\max(X_{1}, X_{2}) with XiX_{i} ~ Exp(1)

“If X and Y are each normal and uncorrelated then they are independent”

False — X ~ N(0,1),ε=±1N(0,1), \varepsilon = \pm1 with probability ½ independent of X,Y=εXX, Y = \varepsilon{}X

“Uncorrelated implies independent”
“Convergence in probability implies almost sure convergence”

False — The typewriter sequence on [0,1]

“Xn→0X_{n} \to 0 almost surely implies E[Xn]→0E[X_{n}] \to 0”
“An irreducible chain with a stationary distribution converges to it”

False — The two-state chain that swaps deterministically (period 2)

“A recurrent chain has a stationary distribution”