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21 votes
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Let $f(x)$ be the continuous probability density function of a random variable $x$, the probability that $a < x \leq b$, is :

  1. $f(b-a)$
  2. $f(b) - f(a)$
  3. $\int\limits_a^b f(x) dx$
  4. $\int\limits_a^b xf (x)dx$
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3 Answers

25 votes
25 votes
Best answer

$A.$ This gives the probability at the point of $b-a$ which is not having any significant w.r.t $a$ and $b.$

$B.$ This gives the difference of the probabilities at $b$ and $a$. Note: This is different from cumulative distribution function $F(b) - F(a).$ Ref: https://en.wikipedia.org/wiki/Cumulative_distribution_function

$C.$ This is Probability Density Function. Ref: https://en.wikipedia.org/wiki/Probability_density_function

$D.$ This is expected value of continuous random variable. Ref: https://en.wikipedia.org/wiki/Expected_value
Answer is $C$.

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4 Comments

For continuous probability density function f(x)

$P(x=a)=\int_a^af(x)dx=0$

Probability at discrete point=0

 So here random variables should be defined as continuous not discrete
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For option (A). $f(b-a)$ gives the probability that the random variable $x$ takes a value near point $(b-a)$

Similarly, for option (B). $f(b) – f(a)$ gives the difference of probabilities that random variable $x$ takes a value near point $b$ and point $a$
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In question f(x) is given probability density function and again the option is probaility density function .how?

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15 votes
15 votes
C should be used if prob density function is given B should be used if prob distribution function is given D must be used to calculate expectation when pdf is given
0 votes
0 votes

f(x) be the continuous probability density function of random variable X.
Then the probablity be area of the corresponding curve i.e.,

Answer:

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