Solve the Following Question.(2 Marks)

Question 12 Marks

Find the probability distribution of number of heads in two tosses of a coin.

Answer

Here the r.v. $X$ takes values $0,1,2$.
Also $S=\{ HH , HT , TH , TT \}$
$
\begin{aligned}
\therefore P(X=0) & =\frac{1}{4}, \\
P(X=1) & =\frac{2}{4}=\frac{1}{2} \\
P(X=2) & =\frac{1}{4},
\end{aligned}
$
$\therefore$ Probability distribution of $X$ is :

$X$	0	1	2
$P ( X =x)$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$

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Question 22 Marks

Find $k$, such that the function
$
P (x)=\left\{\begin{array}{ll}
k\left(\frac{4}{x}\right) & ; \quad x=0,1,2,3,4, k>0 \\
0 ; & \text { otherwise, }
\end{array}\right.
$
is a probability mass function (p.m.f.)

Answer

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Question 32 Marks

A random variable $X \sim N(0,1)$. Find $P(X>0)$ and $P(X<0)$

Answer

Given X ∼ N(0, 1)
∴ μ = 0
$\therefore P(X>\mu)=P(X>0)=\frac{1}{2}$ as the distribution is symmetric about $\mu=0$.

$P(X<\mu)=P(X<0)=\frac{1}{2}$ as the distribution is symmetric about $\mu=0$.

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Question 42 Marks

Find expected value of the random variable $X$ whose probability mass function is :

$X =x$	1	2	3
$P ( X =x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

Answer

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Question 52 Marks

The probability distribution of a discrete random variable $X$ is :

$X =x$	1	2	3	4	5
$P ( X =x)$	$k$	$2k$	$3k$	$4k$	$5k$

Find $( X \leq 4$ )

Answer

Given X is discreate r . v.
∴ P(x) is p.m.f.
p.m.f = ∑ pi = 1
i.e. k + 2k + 3k + 4k + 5k = 1
i.e. 15 k = 1
i.e. $k=\frac{1}{15}$
P(X≤ 4) = P(X =1) + P(X = 2) + P(X = 3) + P(X = 4)
= k + 2k + 3k + 4k
= 10k
$\begin{aligned} & =10 \times \frac{1}{15} \\ & =\frac{2}{3}\end{aligned}$

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Question 62 Marks

The probability mass function $(p.m.f.)$ of $X$ is given below :

$X =x$	$1$	$2$	$3$
$P ( X =x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

Find $E \left( X ^2\right)$.

Answer

$x$	$P(x)$	$xP(x)$	$x^2P(x)$
$1$	$1/5$	$1/5$	$1/5$
$2$	$2/5$	$4/5$	$8/5$
$3$	$2/5$	$6/5$	$18/5$
			$\sum x^2 P(x)=\frac{27}{5}$

$E\left(x^2\right)=\sum x^2 P(x)=\frac{27}{5}$

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Question 72 Marks

Obtain the probability distribution of the number of sixes in two tosses of a fair die.

Answer

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Maths Solve the Following Question.(2 Marks)

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