Let X be a random variable denoting the amount of loss in a business. The moment generating function of X is \(M(t)=(\frac{2}{2-t})^2,\ \ \ \ t< 2.\) If an insurance policy pays 60% of the loss, then the variance of the amount paid by the insurance policy equals __________ (round off to 2 decimal places)

The correct answer is 0.17 to 0.19.(approx)

Let X be a random variable denoting the amount of loss in a business.The moment generating function of X is M(t)=(2/2-t)2,t<2.If an insurance policy pays

Suppose that the weights (in kgs) of six months old babies, monitored at a healthcare facility, have

N(\mu, \sigma^2)

distribution, where

\mu \in \mathbb{R}

and

\sigma > 0

are unknown parameters. Let

X_1, X_2, \ldots, X_9

be a random sample of the weights of such babies. Let

\overline{X} = \frac{1}{9} \sum_{i=1}^{9} X_i

,

S = \sqrt{\frac{1}{8} \sum_{i=1}^{9} (X_i - \overline{X})^2}

and let a 95% confidence interval for

\mu

based on

t

-distribution be of the form

(\overline{X} - h(S), \overline{X} + h(S))

, for an appropriate function

h

of random variable

S

. If the observed values of

\overline{X}

and

S^2

are 9 and 9.5, respectively, then the width of the confidence interval is equal to __________ (round off to 2 decimal places) (You may use

t_{9,0.025} = 2.262, t_{8,0.025} = 2.306, t_{9,0.05} = 1.833, t_{8,0.05} = 1.86

).

IIT JAM MS - 2024
Multivariate Distributions

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Let

X_1, X_2, X_3

be a random sample from a Poisson distribution with mean

\lambda

,

\lambda > 0

. For testing

H_0: \lambda = \frac{1}{8}

against

H_1: \lambda = 1

, a test rejects

H_0

if and only if

X_1 + X_2 + X_3 > 1

. The power of this test is equal to __________ (round off to 2 decimal places).

IIT JAM MS - 2024
Standard Univariate Distributions

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Let

X_1, X_2, \ldots, X_{10}

be a random sample from a

U(-\theta, \theta)

distribution, where

\theta \in (0, \infty)

. Let

X_{(10)} = \max \{ X_1, X_2, \ldots, X_{10} \}

and

X_{(1)} = \min \{ X_1, X_2, \ldots, X_{10} \}

. If the observed values of

X_{(10)}

and

X_{(1)}

are 8 and -10, respectively, then the maximum likelihood estimate of

\theta

is equal to __________ (answer in integer).

IIT JAM MS - 2024
Estimation

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Consider a coin for which the probability of obtaining head in a single toss is

\frac{1}{3}

. Sunita tosses the coin once. If head appears, she receives a random amount of

X

rupees, where

X

has the

\text{Exp}\left( \frac{1}{9} \right)

distribution. If tail appears, she loses a random amount of

Y

rupees, where

Y

has the

\text{Exp}\left( \frac{1}{3} \right)

distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).

IIT JAM MS - 2024
Probability

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Let

X

be a discrete random variable with

P(X \in \{-5, -3, 0, 3, 5\}) = 1

. Suppose that

P(X = -3) = P(X = -5)

,

P(X = 3) = P(X = 5)

and

P(X > 0) = P(X = 0) = P(X < 0)

. Then the value of

12P(X = 3)

is equal to __________ (answer in integer).

IIT JAM MS - 2024
Multivariate Distributions

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Correct Answer: 0.17

Solution and Explanation

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