A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.

(a)

\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 3 } \)

(b)

\( \frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 6 } \)

(c)

\(l,\frac { { l }^{ 2 } }{ 12 } \)

(d)

\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 12 } \)

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

(a)

i + 2n, i = 0,1,2... n

(b)

2i- n, i = 0,1,2... n

(c)

n - i, i = 0,1,2... n

(d)

2i + 2n, i = 0, 1, 2...n

Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are

(a)

50,40

(b)

40,50

(c)

40.75,40

(d)

41,41

On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

(a)

\(\frac { 11 }{ 243 } \)

(b)

\(\frac { 3 }{ 8 } \)

(c)

\(\frac { 1 }{ 243 } \)

(d)

\(\frac { 5 }{ 243 } \)

If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is 

(a)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 6 }\left( \frac { 2 }{ 5 } \right) ^{ 4 }\) 

(b)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 10 }\)

(c)

\(\left( \frac { 10 }{ 5 } \right) { \left( \frac { 3 }{ 5 } \right) }^{ 4 }\left( \frac { 2 }{ 5 } \right) ^{ 6 }\)

(d)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 5 }\left( \frac { 2 }{ 5 } \right) ^{ 5 }\)

Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is

(a)

1

(b)

2

(c)

3

(d)

4

If \(f(x)=\left\{\begin{array}{ll} 2 x & 0 \leq x \leq a \\ 0 & \text { otherwise } \end{array}\right.\) is a probability density function of a random variable, then the value of a is

(a)

1

(b)

2

(c)

3

(d)

4

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

(a)

\(\frac { 57 }{ { 20 }^{ 3 } } \)

(b)

\(\frac { 57 }{ { 20 }^{ 2 } } \)

(c)

\(\frac { { 19 }^{ 3 } }{ { 20 }^{ 3 } } \)

(d)

\(\frac { 57 }{ 20 } \)

IfF(x) is the probability distribution function, then \(F\left( -\infty \right) \) is _____________

(a)

1

(b)

2

(c)

\(\infty \)

(d)

0

If \(f(x)={ Cx }^{ 2 }={ cx }^{ 2 },0 is the p.d.f, of x then c is _____________

(a)

\(\frac { 1 }{ 3 } \)

(b)

\(\frac { 4 }{ 3 } \)

(c)

\(\frac { 8 }{ 3 } \)

(d)

\(\frac { 3 }{ 8 } \)