Answer:
a) X ~ exp ( 10 )
b) E(X) = 0.1 , Var (X) = 0.01
c) P ( X < 0.07 ) = 0.00698
Step-by-step explanation:
Solution:-
- The spike train, used to study neural activity, the given time in between consecutive spikes (ISI) where the firing rate = 10 neurons per seconds.
- Denote a random variable "X"represent a single interspike interval (ISI) having an exponential distribution.
- Where X follows exponential distribution defined by event rate parameter i.e λ.
X ~ Exp ( λ )
- The event rate (λ) is the number of times an event occurs per unit time. Since we are studying a single interspike interval (ISI) - which corresponds to the firing rate. So, event rate (λ) = firing rate = 10 neurons per second. Hence, the distribution is:
X ~ Exp ( 10 )
- The expected value E(X) denotes the amount of time in which a single an event occurs; hence, the time taken for a single neuron.
E(X) = 1 / λ
E(X) = 1 / 10
E(X) = 0.1 s per neuron.
- The variance is the variation in the time taken by a single neuron to be emitted. It is defined as:
Var (X) = 1 / λ^2
Var (X) = 1 / 10^2
Var (X) = 0.01 s^2
- The probability that ISI is less than t = 0.07 seconds: P ( X < t = 0.07 s):
- The cumulative distribution function for exponential variate "X" is:
P ( X < t ) = 1 - e^(-λ*t)
- Plug the values and the determine:
P ( X < 0.07 ) = 1 - e^(-0.1*0.07)
= 1 - 0.99302
= 0.00698
