Question

For each of the following, find the constant c so that p(x) satisfies the condition of being a probability density function of a random variable X:
I. p(x) = c(2/3)^x, x ∈ N
II. p(x) = cx, x ∈ {1,2,3,4,5,6}

Answer 1

Question 18748 Hi, the topic is Probability Density Function of a random variable.

For each of the following, find the constant $c$ so that $p(x)$ satisfies the condition of being a probability density function of a random variable $X$: I. $p(x) = c(2/3)^x$, $x \in N$ II. $p(x) = cx$, $x \in \{1, 2, 3, 4, 5, 6\}$

I've been figuring this for couple of days. Solution. In fact, $p(x)$ is rarely called probability density function. It is just distribution of a r.v., $p(x) = P(X = x)$.

We are to verify two conditions: 1) $p(x) \geq 0$ and $\sum_{x \in N} p(x) = 1$. So,

a) $c > 0$ and $\sum_{x \in N} p(x) = c \sum_{x \in N} (2/3)^x = c \frac{2/3}{1/3} = 2c = 1$, thus $c = 1/2$. Remark: this is the case when $0 \notin N(2$ definitions of natural numbers are used in mathematics.), if $0 \in N$, then $\sum_{x \in N} p(x) = c \sum_{x \in N} (2/3)^x = 3c$, thus $c = 1/3$.

b) $\sum_{x\in \{1,2,3,4,5,6\}}cx = c\frac{6\cdot 7}{2} = 21c$, thus $c = 1 / 21$