List of top Statistics for Economics Questions on Probability theory asked in IIT JAM EN

Let 𝑀 and 𝑁 be events defined on the sample space 𝑆. If 𝑃(𝑀) = \(\frac{1}{3}\) and 𝑃(𝑁^𝑐 ) = \(\frac{1}{4}\) then which one of the following is necessarily CORRECT?

Which of the following functions qualify to be a cumulative density function of a random variable 𝑋 ?

Let the joint probability density function of the random variables 𝑋 and 𝑌 be
\(f(x,y) =   \begin{cases} 1,     & \quad 0<x<1,\,\,\,\,x<y<x+1\\   0,& \quad \ \text{ Otherwise}   \end{cases}\)
Let the marginal density of 𝑋 and 𝑌 be 𝑓_𝑋(𝑥) and 𝑓_𝑌 (𝑦), respectively. Which of the following is/are CORRECT?

Let X∼Normal (0, 1) and 𝑌 = |𝑋|. If the probability density function of 𝑌 is 𝑓𝑌 (y) then for 𝑦>0,\( \sqrt{\frac{\pi}{2}}\) 𝑓𝑌 (𝑦) is

Let 𝑋 ∼ 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 8, 20) and 𝑍 ∼ 𝑈𝑛𝑖𝑓𝑜𝑟𝑚(0, 6) be independent random variables. Let 𝑌 = 𝑋 + 𝑍 and 𝑊 = 𝑋 − 𝑍. Then 𝐶𝑜𝑣(𝑌, 𝑊) is ______ (in integer).

Let 𝑌 ∼ 𝑁𝑜𝑟𝑚𝑎𝑙(3, 1), 𝑊 ∼ 𝑁𝑜𝑟𝑚𝑎𝑙(1, 2) and 𝑋 ∼ 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 (𝑝 = 0.9) where 𝑋 = 1 is success and 𝑋 = 0 is failure. Let 𝑆 = 𝑋𝑌 + (1 − 𝑋)𝑊. Then 𝐸(𝑆) = _________ (round off to 1 decimal place)

From a set comprising of 10 students, four girls 𝐺_𝑖 , 𝑖 = 1, … , 4, and six boys 𝐵_𝑗 , 𝑗 = 1, … , 6, a team of five students is to be formed. The probability that a randomly selected team comprises of 2 girls and 3 boys, with at least one of them to be 𝐵₁ or 𝐵₂ , is equal to

If 𝑋 denotes the sum of the numbers appearing on a throw of two fair six-faced dice then the probability
𝑃( 7 < 𝑋 < 10) = ______ (round off to 2 decimal places).

The portfolio of an investment firm comprises of two risky assets, 𝑆 and 𝑇, whose returns are denoted by random variables 𝑅_𝑠 and 𝑅_𝑇 respectively. The mean, the variance and the covariance of the returns are
𝐸(𝑅_𝑠 ) = 0.08, 𝑉𝑎𝑟(𝑅_𝑠 ) = 0.07, 
𝐸(𝑅_𝑇 ) = 0.05, 𝑉𝑎𝑟(𝑅_𝑇 ) = 0.05, 𝐶𝑜𝑣(𝑅𝑠 , 𝑅_𝑇 ) = 0.04. 
Let 𝑤 be the proportion of assets allotted to 𝑆 so that the return from the portfolio is 𝑅 = 𝑤𝑅_𝑠 + (1 − 𝑤)𝑅_𝑇 . The value of 𝑤 which minimizes 𝑉𝑎𝑟(𝑅) is _____(round off to 2 decimal places)

A number 𝑥 is randomly chosen from the set of the first 100 natural numbers. The probability that 𝑥 satisfies the condition \(𝑥 + \frac{300}{𝑥} > 65\) is ______ (round off to 2 decimal places).

In the hypothesis testing, which of the following defines the size of power of the test?

Which of the following CORRECTLY defines the relationship between the variances of sample means for simple random samples drawn with and without replacement from a normal population?

Suppose that two random samples of sizes n1, and n2 are selected without replacement from two binomial populations with means = \(\mu_1= n_1p_1, \,\,\,\mu_2= n_2p_2,\) and variances \(σ_1 ^2 = n_1p_1q_!, \,\,\,σ_2^2 = n_2p_2q_2\) , respectively. Let the difference of sample proportions $\overline{P_1} $ and $\overline{P_2} $ approximate a normal distribution with mean ($p_1 - p_2$). Then the standard deviation of the difference of sample proportions  $\overline{P_1} $ and $\overline{P_2} $, is

The probability of getting head in a toss of a biased coin is ⅔ . Let the coin be tossed three times independently. Then the probability of getting head in the first two tosses and tail in the final toss is

Which of the following is NOT correct regarding R-squared ($R^2$) and Adjusted R-squared ($\overline{R^2}$)?

Let \(X \sim N(\mu X, \sigma _X ^2)\) and \(Y \sim N(\mu _y, \sigma _y ^2)\) Which of the following is/are NOT correct?

An individual faces an uncertain prospect, where wealth could be Rs. 10 Lakh with probability 0.75 and Rs. 7 Lakh with probability 0.25.
Let the utility function be U(w) = w³. Then the individual will buy full insurance by paying a premium of Rs. _____ Lakh (round off to 2 decimal places).