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(B) We know that the sum of all probabilities in a probability distribution is $1$, so $\sum P(X=x_i) = 1$.$0.1 + 0.15 + 0.3 + 0.25 + k + k = 1$$0.8 +

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$1.94$

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(B) We know that the sum of all probabilities in a probability distribution is $1$, so $\sum P(X=x_i) = 1$.$0.1 + 0.15 + 0.3 + 0.25 + k + k = 1$$0.8 + 2k = 1 \implies 2k = 0.2 \implies k = 0.1$.Now, we calculate the mean $E(X) = \sum x_i P(x_i)$ and $E(X^2) = \sum x_i^2 P(x_i)$:$x_i$$P(x_i)$$x_i P(x_i)$$x_i^2 P(x_i)$$1$$0.1$$0.1$$0.1$$2$$0.15$$0.3$$0.6$$3$$0.3$$0.9$$2.7$$4$$0.25$$1.0$$4.0$$5$$0.1$$0.5$$2.5$$6$$0.1$$0.6$$3.6$Total$1.0$$3.4$$13.5$The variance is given by $Var(X) = E(X^2) - [E(X)]^2$.$Var(X) = 13.5 - (3.4)^2 = 13.5 - 11.56 = 1.94$.

Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:
$X = x$ $1$ $2$ $3$ $4$ $5$ $6$
$P(X = x)$ $0.1$ $0.15$ $0.3$ $0.25$ $k$ $k$

Find the variance of $X$.

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$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$0.1$	$0.15$	$0.3$	$0.25$	$k$	$k$

Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:$X = x$$1$$2$$3$$4$$5$$6$$P(X = x)$$0.1$$0.15$$0.3$$0.25$$k$$k$Find the variance of $X$.