To address the problem, we need more information about the probability density function (PDF) of the random variable $X$. Let's solve the generic aspects and use placeholders where information is missing.

Assume the probability density function is $f_X(x)$, defined over the range $a \leq x \leq b$, with $f_X(x) = 0$ outside this range. The following steps outline the solution:

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(i) Determine the value of the constant $c$ (if part of $f_X(x)$)

The probability density function must satisfy:

$\int_{-\infty}^\infty f_X(x) \, dx = 1$

If $f_X(x)$ is given as $c \cdot g(x)$, where $g(x)$ is a function of $x$, then we solve for $c$ by integrating:

$\int_a^b c \cdot g(x) \, dx = 1 \implies c = \frac{1}{\int_a^b g(x) \, dx}.$

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(ii) Cumulative Distribution Function (CDF), $F_X(x)$

The cumulative distribution function $F_X(x)$ is defined as:

$F_X(x) = P(X \leq x) = \int_{-\infty}^x f_X(t) \, dt$

For $x < a$: $F_X(x) = 0$ (since $f_X(x) = 0$ for $x < a$).

For $x > b$: $F_X(x) = 1$ (since the total probability up to $b$ is 1).

For $a \leq x \leq b$: Evaluate the integral explicitly:

$F_X(x) = \int_a^x f_X(t) \, dt.$

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(iii) Probability $P(X \in [c, d])$:

Using the CDF or directly integrating the PDF:

$P(c \leq X \leq d) = F_X(d) - F_X(c)$

Or:

$P(c \leq X \leq d) = \int_c^d f_X(x) \, dx$

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Additional Notes:

If you upload or specify the actual probability density function $f_X(x)$, I can compute $c$, the CDF, and probabilities for specific intervals as requested.

Would you like to upload the function details or provide more context?

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5 Related Questions:

1. How do you compute the mean and variance of a continuous random variable?
2. How does the cumulative distribution function (CDF) relate to the PDF?
3. Can you provide an example of solving a PDF with a specific range?
4. How can we use the CDF to find probabilities of exceeding certain values?
5. What are the conditions under which a function qualifies as a valid PDF?

Tip:

Always verify that the integral of your PDF over its domain equals 1; this ensures it represents a valid probability distribution.