Question

1. Which of the following statements is true?

 a. The sum of probabilities of a complete set of mutually exclusive events must be 1

b.  Independent events must be mutually exclusive

 c. Mutually exclusive events must be independent

 d. All of the above



2. If the random variable X follows the below distribution, what is the value of a?

f(x)=ax2f(x)=ax2from x = 0 to 1

 4

 3

 2

 1


3. Which of the following statements is true with regards to the probability distribution function f(x) of a random variable X?

 f(x) must be less than 1 for all values of x

 f(x) must be non-negative for all values of x

 f(x) cannot exist for negative values of x

 All of the above



4. If a fair coin is tossed 6 times, what is the expected difference between the number of heads and tails?

 0

 1

 2

 3



​

Answer 1

Answer:

Step-by-step explanation:

a

2

All the above

3

Answer 2

Answer:

1. (a) All the above         2. (d) The value of $a$ is $1$

3. (d) All the above        4. (a) and (c)

Step-by-step explanation:

1.

Occurrence of any one of the event excludes the occurrence of the other event then events A and B are called mutually exclusive events, i.e., if they can not occur simultaneously. More precisely, the sets A and B are disjoint.

Two events are independent (Disjoint) if one event occurred doesn't change the probability of the other event.

Mutually exclusive event ⇔ Disjoint event.

Therefore, option (d) is correct.

2.

Since total probability of an event is $1$.

So, $f(0)+f(1)=1$

⇒ $a(0)^2+a(1)^2=1$

⇒ $0+a=1$

⇒ $a=1$

Therefore, option (d) is correct.

3.

The probability distribution function $f(x)$ of a random variable $x$ is a non-decreasing function.

Also, $0\leq f(x)\leq 1$ for all $x$.

So, all the statements are true.

Therefore, option (d) is correct.

4.

If a coin is tossed six times then the possible difference between the number of heads and the number of tails are as follows:

$6$ heads, $0$ tails = $|6-0|=6$

$5$ heads, $1$ tails = $|5-1|=4$

$4$ heads, $2$ tails = $|4-2|=2$

$3$ heads, $3$ tails = $|3-3|=0$

$2$ heads, $4$ tails = $|2-4|=2$

$1$ heads, $5$ tails = $|1-5|=4$

$0$ heads, $6$ tails = $|0-6|=6$

The difference between heads and tails are $6,4,2$ and $0$.

Therefore, option (a) and (c) are correct.

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