There are ${2}^{6}$numbered cards in a deck among which ${}^{6}{\ufffd}_{\ufffd}$ cards bear the number $\ufffd;\ufffd=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.},6$. From the deck, 10 cards are drawn with replacement. What is the expectation of the sum of their numbers?(Enter the answer correct to 1 decimal accuracy)

*1 point*

*1 point*

An unbiased die is thrown 13 times. After each throw a ‘+’ is recorded for 2 or 5 and ‘-‘ is recorded for 1,3,4 or 6, the signs forming an ordered sequence. To each, except the first and last sign, a random variable ${\ufffd}_{\ufffd};\ufffd=1,2,\mathrm{.}\mathrm{.}\mathrm{.},11$ is associated which takes the value $1$ if both of its neighbouring sign differs from the one between them and $0$ otherwise. If the random variable $\ufffd$ is defined as $\ufffd=3\ufffd+13$ where, $\ufffd={\sum}_{\ufffd=1}^{11}{\ufffd}_{\ufffd}$.

Find the expected value of $\ufffd$.

Find the expected value of $\ufffd$.

*1 point*

An unbiased die is thrown 13 times. After each throw a ‘+’ is recorded for 2 or 5 and ‘-‘ is recorded for 1,3,4 or 6, the signs forming an ordered sequence. To each, except the first and last sign, a random variable ${\ufffd}_{\ufffd};\ufffd=1,2,\mathrm{.}\mathrm{.}\mathrm{.},11$ is associated which takes the value $1$ if both of its neighbouring sign differs from the one between them and $0$ otherwise. If the random variable $\ufffd$ is defined as $\ufffd=3\ufffd+13$ where, $\ufffd={\sum}_{\ufffd=1}^{11}{\ufffd}_{\ufffd}$.

Which of the following statement(s) is/are true?

Which of the following statement(s) is/are true?

*1 point*

Amandeep is in the middle of a bridge of infinite length. He takes the unit step to the right with probability 0.66 and to the left with probability 0.34. Assume that the movements are independent of each other.

Hint: Consider the random variable ${\ufffd}_{\ufffd}$ associated with the ${\ufffd}^{\ufffd\u210e}$ step defined as:

${\ufffd}_{\ufffd}$=$\{\begin{array}{ll}{\textstyle 1}& {\textstyle \text{ifthestepofAmandeepistowardstheright}}\\ {\textstyle -1}& {\textstyle \text{ifthestepofAmandeepistowardstheleft}}\end{array}$

What is the expected distance between the starting point and end point of Amandeep after 5 steps?

Hint: Consider the random variable ${\ufffd}_{\ufffd}$ associated with the ${\ufffd}^{\ufffd\u210e}$ step defined as:

${\ufffd}_{\ufffd}$=$\{\begin{array}{ll}{\textstyle 1}& {\textstyle \text{ifthestepofAmandeepistowardstheright}}\\ {\textstyle -1}& {\textstyle \text{ifthestepofAmandeepistowardstheleft}}\end{array}$

What is the expected distance between the starting point and end point of Amandeep after 5 steps?

*1 point*

Amandeep is in the middle of a bridge of infinite length. He takes the unit step to the right with probability 0.66 and to the left with probability 0.34. Assume that the movements are independent of each other.

Hint: Consider the random variable ${\ufffd}_{\ufffd}$ associated with the ${\ufffd}^{\ufffd\u210e}$ step defined as:

${\ufffd}_{\ufffd}$=$\{\begin{array}{ll}{\textstyle 1}& {\textstyle \text{ifthestepofAmandeepistowardstheright}}\\ {\textstyle -1}& {\textstyle \text{ifthestepofAmandeepistowardstheleft}}\end{array}$

What is the variance distance between the starting point and end point of Amandeep after 5 steps?

Hint: Consider the random variable ${\ufffd}_{\ufffd}$ associated with the ${\ufffd}^{\ufffd\u210e}$ step defined as:

${\ufffd}_{\ufffd}$=$\{\begin{array}{ll}{\textstyle 1}& {\textstyle \text{ifthestepofAmandeepistowardstheright}}\\ {\textstyle -1}& {\textstyle \text{ifthestepofAmandeepistowardstheleft}}\end{array}$

What is the variance distance between the starting point and end point of Amandeep after 5 steps?

A box contains 10 white and 6 black balls. 2 balls are drawn at random without replacement. Find the expected value of the number of white balls drawn. (Enter the answer correct to 2 decimal places)

*1 point*

*1 point*

Rohit wants to open his door with $5$ keys(out of which $1$ will open the door) and tries the keys independently and at random. If unsuccessful keys are eliminated from further selection, then Find the expected number of trials required to open the door.

Hint: Suppose Rohit gets the first success at ${\ufffd}^{\ufffd\u210e}$ trial, i.e., he is unable to open the door in the first $(\ufffd-1)$ trials.

And, P(he gets first success at second trial)=$(1-{\displaystyle \frac{1}{5}})\times {\displaystyle \frac{1}{4}}$

Hint: Suppose Rohit gets the first success at ${\ufffd}^{\ufffd\u210e}$ trial, i.e., he is unable to open the door in the first $(\ufffd-1)$ trials.

And, P(he gets first success at second trial)=$(1-{\displaystyle \frac{1}{5}})\times {\displaystyle \frac{1}{4}}$

$\ufffd$ and $\ufffd$ are independent random variables with means 14 and 20, and variances 2 and 6 respectively. Find the variance of $3\ufffd+6\ufffd$.

*1 point*

Table Q10.1.G. represents the probability mass function of a random variable X

Calculate the value of $\ufffd(2\ufffd+1{)}^{2}$. (Enter the answer correct to 2 decimal places)

$\ufffd$ | -1 | 4 | 5 |
---|---|---|---|

$\ufffd(\ufffd=\ufffd)$ | $\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{9}{18}$ |

*1 point*

*1 point*

Suppose that $\ufffd$ is a random variable for which $\ufffd(\ufffd)=15$ and $\ufffd\ufffd\ufffd(\ufffd)=22$. Find the positive values of $\ufffd$ and $\ufffd$ such that $\ufffd=\ufffd\ufffd-\ufffd$, has expectation $0$ and variance $1$.

Really grateful.

Can you be more specific about the content of your article? After reading it, I still have some doubts. Hope you can help me.

Can you be more specific about the content of your article? After reading it, I still have some doubts. Hope you can help me.