Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 44 - 51 of 72

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 165 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 550.00 or

Question number: 44

» Probability » Expectation

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that E (X - a) 2 is minimized for a = E (X), assuming that the· first 2 moments of X exist.

Explanation

Assume Y = E (X - a) 2

Here we want to minimized Y for a that is

ya=0

aE(Xa)2=0

aE(X22aX+a2… (60 more words) …

Question number: 45

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let X have pdf

f(x)={13;1<x<20;otherwise

Obtain the cdf of Y = X 2.

Explanation

FY(y)=P(Yy)=P(X2y)

=P(yXy)

=yyf(x)dx

=13yydx… (4 more words) …

Question number: 46

» Probability » Convergence » In Distribution

Appeared in Year: 2014

Essay Question▾

Describe in Detail

{X n} is a sequence of independent variables. Show that

XnPXXnDX

where X is a random variable. Is the converse true?

Explanation

The sequence X n converges to X in probability if for any ε > 0

P(XnXε)0asn

The sequence X n converges to the distribution of X as n tends to infinity if

Fn(… (314 more words) …

Question number: 47

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let k > 0 be a constant, and

f(x)={kx(1x);0<x<10;otherwise

Obtain P (X > 0.3).

Explanation

We first find the k value for which the f (x) is purely probability density function that is

01f(x)dx=1

k01x(1x)dx=1

k[x22x… (135 more words) …

Question number: 48

» Probability » Probability of M Events Out of N

Appeared in Year: 2010

Essay Question▾

Describe in Detail

A unbaised die is rolled twice. Let A be the event that the first throw shows a number ≤ 2, and B be the event that the second throw shows at least 5. Show that P (AUB) =5/9.

Explanation

A fair die is rolled twice; the sample space consists of thirty six outcomes. The sample space is

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4),… (174 more words) …

Question number: 49

» Probability » Tchebycheffs Inequality

Appeared in Year: 2009

Essay Question▾

Describe in Detail

Let X ~ BIN (100, 0.2). Compute P [10 ≤ X ≤ 30].

Explanation

X ~ BIN (100, 0.2), where n = 100, p = 0.2, q = 0.8

E (X) =np = 20, Var (X) =npq = 16, σ = 4

P[10X30]=P(XE(X)2σ)

Let… (113 more words) …

Question number: 50

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X have the density function,

f(x)={c(1x2), X <10,otherwise

(i) Find the constant c.

(ii) Find the distribution function.

(iii) Compute P [X > -1/2].

Explanation

(I) To find the value of c using this density function, the integral of density is equal to one by probability definition. The range of X is|X| < 1

X={1Xispositive1Xisn… (224 more words) …

Question number: 51

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Write down the probability mass function of geometric distribution. State and prove its ‘lack of memory property’. Find also the mean and the variance of the distribution.

Explanation

If there are number of trails such that the probability of success is p. So, the probability that there are x failures before the first success is

f(x)=pqx;x=0, 1,2,

This is the probability mass function… (555 more words) …

Sign In