$A$ random variable $X$ has the following probability distribution. Then,$P(2 \leq X < 5) = $ . . . . . .
$X = x$ $1$ $2$ $3$ $4$ $5$ $6$ $P(X = x)$ $K$ $3K$ $5K$ $7K$ $8K$ $K$
| $X = x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
| $P(X = x)$ | $K$ | $3K$ | $5K$ | $7K$ | $8K$ | $K$ |
- A$\frac{3}{5}$
- B$\frac{7}{25}$
- C$\frac{23}{25}$
- D$\frac{24}{25}$
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Let the mean and standard deviation of the probability distribution given by the table below be $\mu$ and $\sigma$ respectively. If $\sigma - \mu = 2$,then find the value of $\sigma$.
$X=x$ $-3$ $0$ $1$ $\alpha$ $P(X=x)$ $\frac{1}{4}$ $K$ $\frac{1}{4}$ $\frac{1}{3}$
| $X=x$ | $-3$ | $0$ | $1$ | $\alpha$ |
| $P(X=x)$ | $\frac{1}{4}$ | $K$ | $\frac{1}{4}$ | $\frac{1}{3}$ |
The following table represents the probability distribution of a random variable $X$ for some $k \in Q$. Find the value of $k$.
$X=x$ $-2$ $-1$ $0$ $1$ $2$ $3$ $P(X=x)$ $0.1$ $k$ $0.2$ $2k$ $0.3$ $k$
| $X=x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
| $P(X=x)$ | $0.1$ | $k$ | $0.2$ | $2k$ | $0.3$ | $k$ |
The probability distribution of a random variable $X$ is given below.
$X = x$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $P(x)$ $0.01$ $0.10$ $0.26$ $0.33$ $0.18$ $0.06$ $K$ $0.04$
Then $P(X \geq 3) - P(X < 6) =$
| $X = x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
| $P(x)$ | $0.01$ | $0.10$ | $0.26$ | $0.33$ | $0.18$ | $0.06$ | $K$ | $0.04$ |
Then $P(X \geq 3) - P(X < 6) =$
The cumulative distribution function (c.d.f.) $F(x)$ of a discrete random variable $X$ is given by the following table:
$X$ $-3$ $-1$ $0$ $1$ $3$ $5$ $7$ $9$ $F(X)$ $0.1$ $0.3$ $0.5$ $0.65$ $0.75$ $0.85$ $0.90$ $1$
Then,find $P[X=3]$.
| $X$ | $-3$ | $-1$ | $0$ | $1$ | $3$ | $5$ | $7$ | $9$ |
|---|---|---|---|---|---|---|---|---|
| $F(X)$ | $0.1$ | $0.3$ | $0.5$ | $0.65$ | $0.75$ | $0.85$ | $0.90$ | $1$ |
Then,find $P[X=3]$.
If $x$ is a random variable with $PMF$ as follows: $P(X = x) = \begin{cases} \frac{5}{16}, & x = 0, 1 \\ \frac{kx}{48}, & x = 2 \\ \frac{1}{4}, & x = 3 \end{cases}$ then find $E(x)$.
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