Learn continuous random variables & graph of probability density function. Practice average value or mean of any probability density function & continuous random variables.

- Consider the height of an adult male chosen at random, or life of a certain randomly chosen battery.
- These quantities are called continuous random variable as their range over an interval of real number. ( often rounded off to nearest integer)
- The cholesterol level of a person is also an example of continuous random variable.
- We are sometimes interested in say what is the probability that a randomly chosen male has height between 65 and 70 inches or blood cholesterol level of a person is between 250 and 350 etc.
- This means that we want to find the probability that a continuous random variable takes values in a certain range.
- Let x denote a continuous random variable then \(P(a\leq X \leq b)\) denotes the value of probability that X lies between 'a' and 'b'.
- Every continuous random variable X has probability density function 'f' that means

\(P(a\leq X\leq b)=\int\limits^b_af(x)dx\)

- In the figure shown the graph of probability density function for X when

X \(\to\) height in inches of adult female in U.S.

Shaded area = probability that height of woman is between 60 and 70 inches.

- Probability density function 'f' will always satisfy

- \(f(x)\geq0\,\forall x\)
- \(\int\limits^\infty_{-\infty}f(x)dx=1\) because probability values always lie in [0,1]

- A function f(x) will qualify for probability density function only if it satisfies the above two criteria.

A The probability that lifetime of bulb is more than 200 hours.

B The probability that lifetime of bulb is less than 400 hours.

C The probability that lifetime of bulb is exactly 600 hours.

D The probability that lifetime of bulb is between 200 and 400 hours.

A \(\alpha=\)72

B \(\alpha=\).048

C \(\alpha=\) .013

D \(\alpha=-\dfrac {1}{2}\)

Let X be a continuous random variable, then \(P(a\leq X\leq b)=\int\limits_a^bf(x) dx\)

where \(f\)is the probability density function of random variable X.

- Suppose we are given the graph of \(f\)which is probability density function of some random variable X, then \(P(a\leq X\leq b)=\int\limits_a^b\,f(x)\,dx=\) Area under \(f(x)\) between x = a and x = b.

\(P(a\leq X\leq b)=\int\limits_a^b\,f(x)\,dx=\)Shaded area

- The mean of a random variable X is the long run average value of random variable X. It can be interpreted as measure of centrality of probability density function.

\(\overline x=\)mean of \(f=\dfrac {\int\limits_{-\infty}^\infty x\, f(x) dx}{\int\limits_{-\infty}^\infty \, f(x) dx}=\int\limits_{-\infty}^\infty x\, f(x) dx\)

\(\because\) \(\int\limits_{-\infty}^\infty \, f(x) dx=1\)

\(\therefore\) \(\overline x=\mu=\int\limits_{-\infty}^{\infty}xf(x)\,dx\) = mean of \(f\)which is the Probability density function of random variable X.

A \(\dfrac {3}{4}\)

B \(\dfrac {1}{2}\)

C \(\dfrac {5}{4}\)

D 6

- Suppose X is a random variable which is the time you wait on hold before an agent of a company you are calling, answers your call, then it is found that.
- \(f(t)= \begin {cases} 0& if&t<0\; (\text {as call cannot be answered before it is made})\\ ce^{-ct}& if& t \geq 0\\ \end {cases}\)
- It is the probability density function of X where c is some constant value (t is the time at which call is answered)
- Also it can be verified that \(\mu=\overline x=\dfrac {1}{c}\) for this distribution.
- So the probability that the call will be answered during first minute is \(\int\limits_0^1 f(t) dt\)
- or it will be answered any time after the 4th minute is \(\int\limits_4^{\infty} f(t) dt\)

A 0.7081

B 0.1045

C 1.3926

D 0.0012